TARCER 1.97

A Mathematica program for the reduction of two-loop propagator integrals

Rolf Mertig
Mertig Research & Consulting
and
Rainer Scharf
Universität Leipzig

18th September, 2003

COMMENTS (RM 18.9.2003): Fixed things for 5.0, eliminated a TVI symmetry (causing infinite recursion) and added
TFI decomposition into products of TAI's (was somehow missing but shows up in, e.g., the 2-loop Higss selfenergy).
Putting TFI back into the Tarcer context, because the fcloops context created trouble ...
Changed default of $Comment to False.

COMMENTS (FO 8.9.2002.): Changed context of TFI to HighEnergyPhysics`fcloops`TFI` and updated other HighEnergyPhysics`* contexts.

COMMENTS (FO 15.3.2001.): Changed PreFactor into Prefactor1 to avoid conflict with FeynArts and FeynCalc.

COMMENTS (FO 15.3.2001.): Changed context of Zeta2 and made definition dependent on not having loaded FeynCalc.

COMMENTS (RM 25.4.): TFISimplify (like TFIRecurse, but without trule[9]).

COMMENTS (RM 23.4.): bug fixed in tlrule[17].

COMMENTS (RM 6.4.):  special TVR relations added to TVR.

COMMENTS (RM 22.3.):  TVI Symmetry added to TVR.

COMMENTS (RM 18.3.): More rules added to $TTable.

COMMENTS (RM 18.2.): More rules added to $TTable.

COMMENTS (RM 21.2.): Included $TTable. added tlrule[25]

COMMENTS (RM 20.2.): $RankLimit = {0,6} ;  Changed TarcerRecurse such that special TJR and TVR (reducing D -> D-2) are used first.

COMMENTS (RM 15.2.): $RankLimit = {0,8} ;  
                                              The following test now works:
                                              
                                                 Print["test 1 :  TFI[D, pp, {0, 0, 0, 1, 3}, {{1, MW}, {1, MZ}, {0, 0}, {1, MZ}, {1,MW}}] = ",
                                                            Timing[t1 =TarcerRecurse[TFI[D, pp, {0, 0, 0, 1, 3}, {{1, MW}, {1, MZ}, {0, 0}, {1, MZ}, {1,MW}}]]
                                                                       ] // InputForm
                                                         ]

                                                  Print["test 2 :  TFI[d, pp, 0, {0, 0, 0, 0, 4},{{1, m1}, {1, m2}, 0, 0, 0}] = ",
                                                           Timing[t2 =TarcerRecurse[TFI[D, pp,  0, {0, 0, 0, 0, 4},{{1, m1}, {1, m2}, 0, 0, 0}]];
                                                                       ] // InputForm
                                                          ]

COMMENTS (RM 13.2.): tlrule[11] and trule[12] made more restrictive: inhibiting thus a recursion if {n5,m5} equals {0,0}.

COMMENTS (RM 12.2.): tlrule[15] made safer (in order to not get   0 :> 0 );
                                             TFIRecurse made safer (to not get into a recursion);
                                                  There is still something wrong with the tlrule's ; since 
                                         TFIRecurse[TFI[D, pp, {0, 0, 0, 1, 3}, {{1, MW}, {1, MZ}, {0, 0}, {1, MZ}, {1,   MW}}]]
                                         doesn't work ..

COMMENTS (RM 5.2.): tlrule[10] slightly changed to cover more cases.

COMMENTS (RM 2.2.):  FunctionExpand added in tlrule[9]. Option TimeConstraint added to TarcerRecurse (because TimeConstrained is buggy under Linux).

COMMENTS (RM 1.2.):  Changed TarcerRecurse. New TLR definition. 
                          default $RankLimit={0,6};

$Remember = False;

COMMENTS (RM 19.1.):   changes are:  tlrule[22] and tlrule[23], minor efficiency changes (Expand3 = Identity) and other small programming optimizations.

COMMENTS (RM 22.1.):   changes are: bug in tlrule[23] fixed, new tlrule[24], new option Factor to TarcerReduce (doesn't change anything by default, but allows for improved Factor-ing).

See also: hep-ph/9801383, published in Comput. Phys. Commun. 111 (1998) 265-273

Table of contents

Preliminaries

Usages

Creating the tarcerij.mx file

Prologue (setting of $RankLimit)

Typesetting rules

The Palettes

OperatorApply

MassDerivative

TarasovT

ApplyTarasovT

Misc

Colors

PQ

PNQ

FactorC

TFI

TJI

TBI

TAI

Definitions and notation

ToArgs

Cayley

Cayley[i]=(Subscript[Δ, i])

CayleyD[i]=(Subscript[Δ, i])/Δ

Utility Functions

MakeFun and friends

Interchange

νexplicit

TA

CheckRecursion

CheckTVIRecursion

CheckTJIRecursion

Recurrence Relations Input and Recursion Function Generation

Input (35)

MakeFun (35)

(1^(+ )) (35  _ 1)

(2^(+ )) (35  _ 2)

(3^(+ )) (35  _ 3)

(4^(+ )) (35  _ 4)

Input (3501)

\(\(MakeFun\ \((3501)\)\ Δ\)\(=\)\(0\)\(\ \)\)((p^2) = (m^2))

Input (3502)

\(\(MakeFun\ \((3502)\)\ Δ\)\(=\)\(0\)\(\ \)\)((p^2) = (m^2))

\(\(MakeFun\ \((3503)\)\ Δ\)\(=\)\(0\)\(\ \)\)((p^2) = (m^2))

\(\(MakeFun\ \((3504)\)\ Δ\)\(=\)\(0\)\(\ \)\)((p^2) = (m^2))

Input (41)

MakeFun (41)

Input (43)

Input (44)

Input (45)

MakeFun (43) and (44)

MakeFun (50)

Input (51)

MakeFun (51)

Input (52)

MakeFun (52)

MakeFun (53)

Input (54)

MakeFun (54)

Input (55)

MakeFun (55)

Input (56)

MakeFun (56)

Input (59)

MakeFun (59)

Input (60)

MakeFun (60)

Input (61)

MakeFun (61)

MakeFun (6153)

Input (62)

MakeFun (62)

Input (63)

MakeFun (63)

MakeFun (6552)

MakeFun (6552)

MakeFun (6553)

Input (66)

MakeFun (66)

Input (alternative67)

Input (67)

MakeFun (67)

MakeFun (6713)

Input (68)

MakeFun (68)

MakeFun (6813)

Input (70)

MakeFun (70)

Input (71)

MakeFun (71)

MakeFun (7123)

MakeFun (7113)

Input (78)

MakeFun (78)

Input (80)

MakeFun (80)

Input (81)

MakeFun (81)

MakeFun (8123)

MakeFun (8113)

Input (82)

MakeFun (82)

MakeFun (8213)

MakeFun (8212)

Input (83)

MakeFun (83)

Input (87)

(88)

(89)

MakeFun (87)

Input (90)

MakeFun (90)

Input (91)

MakeFun (911)

MakeFun (912)

MakeFun (913)

MakeFun (914)

Input (92)

Input (93)

MakeFun (93)

Input (94)

MakeFun (94)

MakeFun (9401)

MakeFun (9402)

Input (95)

MakeFun (95)

Input (96)

MakeFun (96)

Input (97)

MakeFun (97)

Input (98)

MakeFun (98)

Input (99)

MakeFun (99)

MakeFun (9912)

MakeFun (9913)

Input (100)

MakeFun (100)

MakeFun (1002)

MakeFun (1003)

MakeFun (1004)

MakeFun (1005)

MakeFun (1006)

Input (150)

MakeFun (150)

Input (151)

MakeFun (151)

Input (152)

MakeFun (152)

Input (153)

MakeFun (153)

Input (200)

MakeFun (200)

MakeFun (201)

MakeFun (202)

MakeFun (203)

MakeFun (204)

MakeFun (205)

Input (1xxxx)

five -> four

four -> three

three -> two

two -> one

TarcerRecurse and TFIRecurse

TFIRecurse

T*IS

TComment

TarcerRecurse

TarcerExpand

Construct T's

$BasisIntegrals

Save

Preliminaries

This is free software.
You may use it, distribute it, change it, improve it, etc..
If you have Mathematica questions, ask them to support@wolfram.com or to your local Mathematica reseller.

We consider the following type of integrals, where a,b,u,v,r,s,t are non-negative integers.

TFI[d, 
        (p^2), Δp, {a, b}, {u, v, r, s, t}, {{(Subscript[ν, 1]), (Subscript[m, 1])}, {(Subscript[ν, 2]),(Subscript[m, 2])},{(Subscript[ν, 3]), (Subscript[m, 3])},{(Subscript[ν, 4]), (Subscript[m, 4])},{(Subscript[ν, 5]), (Subscript[m, 5])}}] = 

(1/(π^d))∫ ∫ ((\(\(d\^d\) k\_1\ \(d\^d\) 
                k\_2\ \(\((Δ·
                        k\_1)\)\^a\) \((Δ·k\_1)\)\^b\)((((Subscript[k, 1]^2)))^u)((((Subscript[k, 2]^2)))^v)(((p·(Subscript[k, 1])))^r)(((p·
                    (Subscript[k, 2])))^s)((((Subscript[k, 1])·
                    (Subscript[k, 2])))^t))/(\(\(\(\(\(\(\([\)\(k\_1\%2 - 
                                        m\_1\%2\)\(]\)\)\^ν\_1\)[
                                  k\_2\%2 - 
                                    m\_2\%2]\^ν\_2\)[\((k\_1 - p)\)\^2 - 
                              m\_3\%2]\^ν\_3\)[\((k\_2 - p)\)\^2 - 
                        m\_4\%2]\^ν\_4\)[\((k\_1 - k\_2)\)\^2 - 
                  m\_5\%2]\^ν\_5\)))

Putting the        $Post = (#;)&;       instruction after   
Unset[$Post]   suppresses all output cells.
Putting Unset[$Post] behind  $Post=(#:)&;
will produce a lot of typsetted reduction formulae.

Unset[$Post];

$Post  =  (#; ) &;

If[$Notebooks =!= True, $Post  =  (#; ) &];

$HistoryLength = 0;

This determines the directory into which the resulting file tarcer.mx is written.

If[$Notebooks, 
      SetDirectory[
        ToFileName[
          First["FileName" /. 
              NotebookInformation[InputNotebook[]]]]]]

Somehow Mathematica crashes on my (R.M.) good old NeXTStation if I don't open all cells first ...; this opens all cells automatically:

If[$System === "NeXT", 
      SelectionMove[EvaluationNotebook[], All, Notebook]; 
      FrontEndExecute[{FrontEnd`FrontEndToken[
            EvaluationNotebook[], "SelectionOpenAllGroups"]}]]

Off[General::spell];

Off[General::spell1];

Creating the tarcerij.mx file

Running this notebook will produce a tarcerij.mx file where ij will be replaced by 
the two number set by the variable $RankLimit (default 33). 
In the same directory as this notebook which can be loaded  into Mathematica by    <<tarcer33.mx,
or, when running in Terminal mode, on Unix systems, already when starting Mathematica, as 
myshellprompt>  math -initfile tarcer33.mx

Usages

BeginPackage["HighEnergyPhysics`Tarcer`"];

TarcerRecurse::usage = "TarcerRecurse[expr] applies recurrence \
relations for all TFI's (and TVI, TJI, TBI, TAI) in expr. For evaluation and \
Laurent expansion of the resulting basis integrals use TarcerExpand.";

TarcerExpand::usage = "TarcerExpand[expr, d -> (4+e), order] or \
TarcerExpand[epxr, d -> (4-e), order] or TarcerExpand[expr, d -> (4 - 2 \
om)substitutes integrals as specified by the option TarcerRules, and does a \
Laurent expansion in the variable around 0 at the right hand side of the \
second argument (i.e., you can use whatever convention you like ...). The \
same effect as TarcerExpand[TarcerRecurse[expr], ...] can be achieved by \
setting the option TarcerRecurse of TarcerExpand to True. TarcerExpand[expr, \
d -> (4+e)] is equivalent to TarcerExpand[expr, d -> (4+e), 0].";

TarcerReduce::usage = "TarcerReduce is an option to TarcerExpand. It \
should be set to a list of basic integrals.";

Cayley::usage = "Cayley[m1,m2,m3,m4,m5,Sqrt[pp]] is the Cayley \
determinant.";

Cayleyu::usage = "Cayleyu are the sub-Cayley-determinants as \
described in Tarasov's paper.";

CayleyD::usage = "CayleyD are the sub-Cayley-determinants as \
described in Tarasov's paper.";

Prefactor1::usage = "Prefactor1[expr] may be used to prohibit \
TarcerExpand from Taylor-expanding expr.";

SEpsilon::usage = "Sepsilon[d] is an abbreviation for Exp[(d-4)/2 \
EulerGamma].";

TFI::"usage" =
"TFI[d, pp, {{n1,m1},{n2,m2},{n3,m3},{n4,m4},{n5,m5}}] is
the 2-loop d-dimensional integral
1/( (q1^2 - m1^2)^n1  (q2^2 - m2^2)^n2 ((q1-p)^2 - m3^2)^n3 *
    ((q2-p)^2 - m4^2)^n4  ((q1-q2)^2 - m5^2)^n5 ) .

 TFI[d, pp, {x,y,z,v,w}, {{n1,m1},{n2,m2},{n3,m3},{n4,m4},{n5,m5}}]
has as additional factors in the numerator (q1^2)^x*(q2^2)^y*(q1.p)^z*
(q2.p)^v*(q1.q2)^w .

 TFI[d, pp, dp, {a,b}, {{n1,m1},{n2,m2},{n3,m3},{n4,m4},{n5,m5}}]
has as additional factors in the numerator  (OPEDelta.q1)^a * (OPEDelta.q2)^b;
dp is (OPEDelta.p).";

TFIRecurse::usage = "TFIRecurse[expr] applies the recursion algorithm \
implemented in TARCER.nb (following Tarasov) to TFI's (and TVI's, TJI's, \
TBI's, TAI's in expr.";

TFISimplify::usage = "TFISimplify[expr] applies the recursion \
algorithm implemented in TARCER.nb (following Tarasov) to TFI's (and TVI's, \
TJI's, TBI's, TAI's in expr, except k1.k2.";

TVI::usage = "TVI[d,pp,{ {n1,m1},{n2,m2},{n3,m3},{n4,m4}}] is the \
massive 2-loop integral with four propagators in d dimensions. ";

TJI::usage = "TJI[d, pp, { {n1,m1},{n2,m2},{n3,m3} }] is the massive \
2-loop integral with three propagators in d dimensions. ";

TKI::usage = "TKI[d, 0, { {n1,m1},{n2,m2},{n3,m3} }] is the bubble \
massive 2-loop selfenergy type integral with three propagators in d \
dimensions. ";

TBI::usage = "TBI[d, pp, { {n1,m1},{n2,m2} }] is the massive 1-loop \
selfenergy type integral with two propagators in d dimensions.";

TAI::usage = "TAI[d, pp, { {n1,m1} }] is the massive 1-loop \
selfenergy type integral with one propagator in d dimensions.";

TFR::usage = "TFR is similar in syntax to TFI but applies once the \
recurrence relations derived for TFI.";

TVR::usage = "TVR is similar in syntax to TVI but applies once the \
recurrence relations derived for TVI.";

TJR::usage = "TJR is similar in syntax to TJI but applies once the \
recurrence relations derived for TJI.";

TBR::usage = "TBR is similar in syntax to TBI but applies once the \
recurrence relations derived for TBI.";

TAR::usage = "TAR is similar in syntax to TAI but applies once the \
recurrence relations derived for TAI.";

$RankLimit::usage = "$RankLimit is the list of integers specifying \
the highest possible sums of exponents of scalar products; i.e., {a+b, r+s} \
means that at most (Delta.k1)^a * (Delta.k2)^2 * (p.k1)^r * (p.k2)^s in the \
numerator will be calculated (a,b,r,s have to be non-negative integares). \
$RankLimit has to be set in the TARCER.nb notebook. In the tarcer.mx file \
this is not possible anymore.";

\($TarcerRecursed::usage = "If $Comment is set to True all integrals
		which are reduced are collected in the global variable $TarcerRecured."; \)

\($BasisIntegrals::usage = "$BasisIntegrals is a list of some basis integrals. It is the default setting of TarcerReduce of TarcerExpand."; \)

$Comment::usage = "$Comment can be set to True or False. If set to \
False no comments are generated.";

$CommentNotebook::usage = "$Comment can be set to True or False. \
Setting it to True will generate a separate notebook (if and only if $Comment \
is True) where the used recursion relations are displayed.";

\($TTable::usage = "$TTable is a list of precomputed reductions. $TTable is the default setting of the option Table of TarcerRecurse. "; \)

Begin["`Private`"];

Expand3 = Identity;

Prologue (setting of $RankLimit)

This sets the limit up to which the T-operators are constructed automatically, i.e., $RankLimit = {a+b, r+s} , concerning: (((Δ·
              (Subscript[k, 1])))^a) (((Δ·(Subscript[k, 2])))^b) (((p·(Subscript[k, 1])))^r)(((p·(Subscript[k, 2])))^s)

$RankLimit = {2, 6};

$RankLimit = {0, 6};

Check recursion relations by changing False to True here:

$CheckRecursion = False;

Change False to True for getting the input palettes for entering new recursion relations.

$PutPalettes = False;

Notice:  Instead of using shift operators  \(1\^+\),\(2\^+\) etc. the ν-indices are colored blue; instead of   \(1\^-\),\(2\^-\) etc. the ν-indices are colored red.

Typesetting rules

atime = AbsoluteTime[];

SEpsilon /: MakeBoxes[SEpsilon[d_], fmt_] := 
      InterpretationBox @@ {StyleBox[
            SubscriptBox["S", First[Variables[d]]], 
            FontWeight -> "Bold"], SEpsilon[d], 
          Editable -> False}

MakeBoxes[SEpsilon[d_]^n_, fmt_] := 
      InterpretationBox @@ {StyleBox[
            SubsuperscriptBox["S", First[Variables[d]], n], 
            FontWeight -> "Bold"], SEpsilon[d], 
          Editable -> False}

Prefactor1 /: MakeBoxes[Prefactor1[d_], fmt_] := 
      InterpretationBox @@ {MakeBoxes[d, fmt], Prefactor1[d], 
          Editable -> False}

Don't mess with the InterpretationBox's !! (important for the palette and OperatorApply).

MakeBoxes[Schiebe[i_, pm_String] , fmt_] := 
      InterpretationBox[
        StyleBox[SuperscriptBox[i, pm], FontWeight -> "Bold"], 
        Schiebe[i, pm], Editable -> True]

CayleyD /: MakeBoxes[CayleyD[i_Integer][args_], fmt_] := 
      FractionBox[
        SubscriptBox["Δ",  
          i], "Δ"];

CayleyD /: MakeBoxes[CayleyD[i_Integer, j_Integer][args_], fmt_] := 
      FractionBox[
        SubscriptBox["Δ", 
          10 i + j], "Δ"];

CayleyD /: 
      MakeBoxes[CayleyD[i_Integer, j_Integer, k_Integer][args_], fmt_] := 
      FractionBox[
        SubscriptBox["Δ", 
          100  i + 10  j + k], "Δ"];

Cayley /: MakeBoxes[Cayley[i_Integer][args_], fmt_] := 
      SubscriptBox["Δ",  i];

Cayley /: MakeBoxes[Cayley[i_Integer, j_Integer][args_], fmt_] := 
      SubscriptBox["Δ", 10 i + j];

Cayley /: 
      MakeBoxes[Cayley[i_Integer, j_Integer, k_Integer][args_], fmt_] := 
      SubscriptBox["Δ", 100  i + 10  j + k];

Cayleyu /: 
      MakeBoxes[Cayleyu[i_Integer, j_Integer, k_Integer][args_], fmt_] := 
      SubscriptBox["u", 100  i + 10  j + k];

MakeBoxes[PP, fm_] := InterpretationBox[SuperscriptBox[p, 2], PP]; 
MakeBoxes[m1^(p_), _] := InterpretationBox[SubsuperscriptBox[m, 1, p], m1^p]; 
MakeBoxes[m2^(p_), _] := InterpretationBox[SubsuperscriptBox[m, 2, p], m2^p]; 
MakeBoxes[m3^(p_), _] := InterpretationBox[SubsuperscriptBox[m, 3, p], m3^p]; 
MakeBoxes[m4^(p_), _] := InterpretationBox[SubsuperscriptBox[m, 4, p], m4^p]; 
MakeBoxes[m5^(p_), _] := InterpretationBox[SubsuperscriptBox[m, 5, p], m5^p]; 
m6 = Sqrt[PP];

MakeBoxes[m1, _] := 
      InterpretationBox[SubscriptBox[m, 1], m1];

MakeBoxes[m2, _] := 
      InterpretationBox[SubscriptBox[m, 2], m2];

MakeBoxes[m3, _] := 
      InterpretationBox[SubscriptBox[m, 3], m3];

MakeBoxes[m4, _] := 
      InterpretationBox[SubscriptBox[m, 4], m4];

MakeBoxes[m5, _] := 
      InterpretationBox[SubscriptBox[m, 5], m5];

mbt[z_] := ToBoxes[z, TraditionalForm];

redblue[z_ /; Head[z] =!= Plus] := mbt[z];

redblue[z_ - 1] := 
      StyleBox[mbt[z], FontColor -> RGBColor[1, 0, 0]];

redblue[z_Subscript - 2] := 
      StyleBox[SubscriptBox[
          OverscriptBox[
            z[[1]], "_"], 
          z[[2]]], 
        FontColor -> RGBColor[1, 0, 0]];

redblue[z_ - 2] := 
      StyleBox[OverscriptBox[mbt[z], "_"], 
        FontColor -> RGBColor[1, 0, 0]];

redblue[z_ - 3] := 
      StyleBox[UnderscriptBox[OverscriptBox[mbt[z], "_"], "_"], 
        FontColor -> RGBColor[1, 0, 0]];

redblue[z_ + 1] := 
      StyleBox[mbt[z], FontColor -> RGBColor[0, 0, 1]];

redblue[z_ + 2] := 
      StyleBox[OverscriptBox[mbt[z], "_"], 
        FontColor -> RGBColor[0, 0, 1]];

redblue[z_Subscript + 2] := 
      StyleBox[SubscriptBox[
          OverscriptBox[
            z[[1]], "_"], 
          z[[2]]], 
        FontColor -> RGBColor[0, 0, 1]];

\(TFI /: 
      MakeBoxes[TFI[d_, pp_, {ur__}, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["F", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[d, TraditionalForm], ")", " ", 
                    ur}]], 
		TFI[d, pp, {ur}, {den}], 
              Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

\(TFI /: 
      MakeBoxes[TFI[d_, pp_, dp_, any__, {den__}], 
        fmt_] := 
	\(\((InterpretationBox @@ {SubsuperscriptBox[
                  StyleBox["F", SingleLetterItalics -> False, 
                    FontWeight -> "Bold"], 
                  RowBox @@ {\((redblue /@ {den})\)}, 
                  RowBox[{"(", 
                      ToBoxes[d, TraditionalForm], ")", " ", 
                      Apply[Sequence, Flatten[{any}]]}]], 
		TFI[d, pp, 
                  dp, any, {den}], Editable -> True})\) /; 
          MemberQ[{StandardForm, TraditionalForm}, fmt]\) /; 
        Head[dp] =!= List; \)

\(TFI /: 
      MakeBoxes[TFI[dpp__, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["F", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[First[{dpp}], 
                      TraditionalForm], ")"}]], 
		TFI[dpp, {den}], 
              Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt] && 
          Head[Last[{dpp}]] =!= List; \)

\(TVI /: 
      MakeBoxes[TVI[dpp__, {den__}], 
        fmt_] := 
\((	\((InterpretationBox @@ {SubsuperscriptBox[
                    StyleBox["V", SingleLetterItalics -> False, 
                      FontWeight -> "Bold"], 
                    RowBox @@ {\((redblue /@ {den})\)}, 
                    RowBox[{"(", 
                        ToBoxes[{dpp}[\([\)\(1\)\(]\)], 
                          TraditionalForm], ")"}]], 
	\(Hold[TVI]\)[
                    dpp, {den}], Editable -> True})\) /. Hold[TVI] -> TVI)\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt] && 
          Head[Last[{dpp}]] =!= List; \)

{TVI[d, {α, β, γ, δ}], 
      TVI[d, {α + 1, β, γ, δ}], 
      TVI[d, {α - 1, β, γ, δ}]}

\(TJI /: 
      MakeBoxes[TJI[d_, 0, {den__}], 
        fmt_] := 	\((
\((\((InterpretationBox @@ {SubsuperscriptBox[
                        StyleBox["K", SingleLetterItalics -> False, 
                          FontWeight -> "Bold"], 
                        RowBox @@ {\((redblue /@ {den})\)}, 
                        RowBox[{"(", 
                            ToBoxes[d, 
                              TraditionalForm], ")"}]], 
		\(Hold[
                          TJI]\)[d, 0, {den}], Editable -> True})\) /. 
                Hold[TJI] -> TJI)\) /; 
            MemberQ[{StandardForm, TraditionalForm}, fmt]
	)\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]\)

\(TJI /: 
      MakeBoxes[TJI[dpp__, {den__}], 
        fmt_] := 	\((\((InterpretationBox @@ {SubsuperscriptBox[
                    StyleBox["J", SingleLetterItalics -> False, 
                      FontWeight -> "Bold"], 
                    RowBox @@ {\((redblue /@ {den})\)}, 
                    RowBox[{"(", 
                        ToBoxes[{dpp}[\([\)\(1\)\(]\)], 
                          TraditionalForm], ")"}]], 
		\(Hold[TJI]\)[
                    dpp, {den}], Editable -> True})\) /. Hold[TJI] -> TJI)\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]\)

\(TKI /: 
      MakeBoxes[TKI[d_, pp___, {den__}], 
        fmt_] := 
	\((SubsuperscriptBox[
            StyleBox["K", SingleLetterItalics -> False, 
              FontWeight -> "Bold"], RowBox @@ {\((redblue /@ {den})\)}, 
            RowBox[{"(", ToBoxes[d, TraditionalForm], ")"}]])\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

\(TKI /: 
      MakeBoxes[TKI[dpp__, {den__}], 
        fmt_] := 	\((\((InterpretationBox @@ {SubsuperscriptBox[
                    StyleBox["K", SingleLetterItalics -> False, 
                      FontWeight -> "Bold"], 
                    RowBox @@ {\((redblue /@ {den})\)}, 
                    RowBox[{"(", 
                        ToBoxes[{dpp}[\([\)\(1\)\(]\)], 
                          TraditionalForm], ")"}]], 
		\(Hold[TKI]\)[
                    dpp, {den}], Editable -> True})\) /. Hold[TKI] -> TKI)\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]\)

{TKI[d, {α, β, γ}], 
      TKI[d, {α + 1, β, γ}], 
      TKI[d, {α - 1, β, γ}]}

\(TBI /: 
      MakeBoxes[TBI[d_, pp___, {den__}], 
        fmt_] := 
	\((SubsuperscriptBox[
            StyleBox["B", SingleLetterItalics -> False, 
              FontWeight -> "Bold"], RowBox @@ {\((redblue /@ {den})\)}, 
            RowBox[{"(", ToBoxes[d, TraditionalForm], ")"}]])\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

\(TBI /: 
      MakeBoxes[TBI[d_, pp___, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["B", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[d, TraditionalForm], ")"}]], 
		TBI[d, 
                pp, {den}], Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

{TBI[d, {α, β}], TBI[d, {α + 1, β}], 
      TBI[d, {α - 1, β}]}

\(TAI /: 
      MakeBoxes[TAI[d_, pp___, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["A", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[d, TraditionalForm], ")"}]], 
		TAI[
                d, {den}], Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

{TAI[d, {α}], TAI[d, {α + 1}], 
      TAI[d, {α - 1}]}

T*IC are just like T*I , but used only for commentary in the 
recursion (TComment).

\(TFIC /: 
      MakeBoxes[TFIC[d_, pp_, {0, 0, i_, 0, 0}, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["F", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[d, TraditionalForm], ")", " ", 
                    i, "0"}]], 
		TFIC[d, pp, {0, 0, i, 0, 0}, {den}], 
              Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

\(TFIC /: 
      MakeBoxes[TFIC[d_, pp_, {0, 0, 0, i_, 0}, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["F", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[d, TraditionalForm], ")", " ", "0", 
                    i}]], 
		TFIC[d, pp, {0, 0, 0, i, 0}, {den}], 
              Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

\(TFIC /: 
      MakeBoxes[TFIC[dpp__, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["F", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[First[{dpp}], 
                      TraditionalForm], ")"}]], 
		TFIC[dpp, {den}], 
              Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt] && 
          Head[Last[{dpp}]] =!= List; \)

\(TVIC /: 
      MakeBoxes[TVIC[dpp__, {den__}], 
        fmt_] := 
\((	\((InterpretationBox @@ {SubsuperscriptBox[
                    StyleBox["V", SingleLetterItalics -> False, 
                      FontWeight -> "Bold"], 
                    RowBox @@ {\((redblue /@ {den})\)}, 
                    RowBox[{"(", 
                        ToBoxes[{dpp}[\([\)\(1\)\(]\)], 
                          TraditionalForm], ")"}]], 
	\(Hold[TVIC]\)[
                    dpp, {den}], Editable -> True})\) /. 
            Hold[TVIC] -> TVIC)\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt] && 
          Head[Last[{dpp}]] =!= List; \)

\(TJIC /: 
      MakeBoxes[TJIC[d_, 0, {den__}], 
        fmt_] := 	\((
\((\((InterpretationBox @@ {SubsuperscriptBox[
                        StyleBox["K", SingleLetterItalics -> False, 
                          FontWeight -> "Bold"], 
                        RowBox @@ {\((redblue /@ {den})\)}, 
                        RowBox[{"(", 
                            ToBoxes[d, 
                              TraditionalForm], ")"}]], 
		\(Hold[
                          TJIC]\)[d, 0, {den}], Editable -> True})\) /. 
                Hold[TJIC] -> TJIC)\) /; 
            MemberQ[{StandardForm, TraditionalForm}, fmt]
	)\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]\)

\(TJIC /: 
      MakeBoxes[TJIC[dpp__, {den__}], 
        fmt_] := 	\((\((InterpretationBox @@ {SubsuperscriptBox[
                    StyleBox["J", SingleLetterItalics -> False, 
                      FontWeight -> "Bold"], 
                    RowBox @@ {\((redblue /@ {den})\)}, 
                    RowBox[{"(", 
                        ToBoxes[{dpp}[\([\)\(1\)\(]\)], 
                          TraditionalForm], ")"}]], 
		\(Hold[TJIC]\)[
                    dpp, {den}], Editable -> True})\) /. 
            Hold[TJIC] -> TJIC)\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]\)

\(TBIC /: 
      MakeBoxes[TBIC[d_, pp___, {den__}], 
        fmt_] := 
	\((SubsuperscriptBox[
            StyleBox["B", SingleLetterItalics -> False, 
              FontWeight -> "Bold"], RowBox @@ {\((redblue /@ {den})\)}, 
            RowBox[{"(", ToBoxes[d, TraditionalForm], ")"}]])\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

\(TBIC /: 
      MakeBoxes[TBIC[d_, pp___, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["B", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[d, TraditionalForm], ")"}]], 
		TBIC[d, 
                pp, {den}], Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

\(TAIC /: 
      MakeBoxes[TAIC[d_, pp___, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["A", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[d, TraditionalForm], ")"}]], 
		TAIC[
                d, {den}], Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

\(TFR /: 
      MakeBoxes[TFR[d_, pp_, {0, 0, i_, 0, 0}, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["F", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[d, TraditionalForm], ")", " ", 
                    i, "0"}]], 
		TFR[d, pp, {0, 0, i, 0, 0}, {den}], 
              Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

\(TFR /: 
      MakeBoxes[TFR[d_, pp_, {0, 0, 0, i_, 0}, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["F", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[d, TraditionalForm], ")", " ", "0", 
                    i}]], 
		TFR[d, pp, {0, 0, 0, i, 0}, {den}], 
              Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

\(TFR /: 
      MakeBoxes[TFR[dpp__, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["F", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[First[{dpp}], 
                      TraditionalForm], ")"}]], 
		TFR[dpp, {den}], 
              Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt] && 
          Head[Last[{dpp}]] =!= List; \)

\(TVR /: 
      MakeBoxes[TVR[dpp__, {den__}], 
        fmt_] := 
\((	\((InterpretationBox @@ {SubsuperscriptBox[
                    StyleBox["V", SingleLetterItalics -> False, 
                      FontWeight -> "Bold"], 
                    RowBox @@ {\((redblue /@ {den})\)}, 
                    RowBox[{"(", 
                        ToBoxes[{dpp}[\([\)\(1\)\(]\)], 
                          TraditionalForm], ")"}]], 
	\(Hold[TVR]\)[
                    dpp, {den}], Editable -> True})\) /. Hold[TVR] -> TVR)\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt] && 
          Head[Last[{dpp}]] =!= List; \)

\(TJR /: 
      MakeBoxes[TJR[d_, 0, {den__}], 
        fmt_] := 	\((
\((\((InterpretationBox @@ {SubsuperscriptBox[
                        StyleBox["K", SingleLetterItalics -> False, 
                          FontWeight -> "Bold"], 
                        RowBox @@ {\((redblue /@ {den})\)}, 
                        RowBox[{"(", 
                            ToBoxes[d, 
                              TraditionalForm], ")"}]], 
		\(Hold[
                          TJR]\)[d, 0, {den}], Editable -> True})\) /. 
                Hold[TJR] -> TJR)\) /; 
            MemberQ[{StandardForm, TraditionalForm}, fmt]
	)\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]\)

\(TJR /: 
      MakeBoxes[TJR[dpp__, {den__}], 
        fmt_] := 	\((\((InterpretationBox @@ {SubsuperscriptBox[
                    StyleBox["J", SingleLetterItalics -> False, 
                      FontWeight -> "Bold"], 
                    RowBox @@ {\((redblue /@ {den})\)}, 
                    RowBox[{"(", 
                        ToBoxes[{dpp}[\([\)\(1\)\(]\)], 
                          TraditionalForm], ")"}]], 
		\(Hold[TJR]\)[
                    dpp, {den}], Editable -> True})\) /. Hold[TJR] -> TJR)\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]\)

\(TBR /: 
      MakeBoxes[TBR[d_, pp___, {den__}], 
        fmt_] := 
	\((SubsuperscriptBox[
            StyleBox["B", SingleLetterItalics -> False, 
              FontWeight -> "Bold"], RowBox @@ {\((redblue /@ {den})\)}, 
            RowBox[{"(", ToBoxes[d, TraditionalForm], ")"}]])\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

\(TBR /: 
      MakeBoxes[TBR[d_, pp___, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["B", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[d, TraditionalForm], ")"}]], 
		TBR[d, 
                pp, {den}], Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

\(TAR /: 
      MakeBoxes[TAR[d_, pp___, {den__}], 
        fmt_] := 
	\((InterpretationBox @@ {SubsuperscriptBox[
                StyleBox["A", SingleLetterItalics -> False, 
                  FontWeight -> "Bold"], 
                RowBox @@ {\((redblue /@ {den})\)}, 
                RowBox[{"(", 
                    ToBoxes[d, TraditionalForm], ")"}]], 
		TAR[
                d, {den}], Editable -> True})\) /; 
        MemberQ[{StandardForm, TraditionalForm}, fmt]; \)

n1 /: MakeBoxes[n1, TraditionalForm] := 
      SubscriptBox["ν", 1];

n2 /: MakeBoxes[n2, TraditionalForm] := 
      SubscriptBox["ν", 2];

n3 /: MakeBoxes[n3, TraditionalForm] := 
      SubscriptBox["ν", 3];

n4 /: MakeBoxes[n4, TraditionalForm] := 
      SubscriptBox["ν", 4];

n5 /: MakeBoxes[n5, TraditionalForm] := 
      SubscriptBox["ν", 5];

MakeBoxes[ParD[i_]^j_, TraditionalForm] := 
      SubsuperscriptBox["∂", i, j];

MakeBoxes[ParD[i_], TraditionalForm] ^= 
      SubscriptBox["∂", i];

MakeBoxes[DPlus, TraditionalForm] := 
      StyleBox[SuperscriptBox["d", "+"], 
        FontWeight -> "Bold", 
        SingleLetterItalics -> False];

MakeBoxes[TpD[_, b_], TraditionalForm] := 
      MakeBoxes[b, TraditionalForm]

The Palettes

\(If[$PutPalettes === 
        True, 
	\(\(tobut[a_]\  := \ 
          ButtonBox[ToBoxes[a, StandardForm], \ 
            ButtonStyle -> "Paste"]; \)\(
\)\(\ \)\(If[
          Head[pal1] === NotebookObject, 
	
          NotebookClose[pal1]]; \)\(
\)\(	\)\(pal1 = 
          NotebookPut[
            Notebook[{Cell[
                  BoxData[
                    GridBox[
							Join[
                        Transpose@{
									Table[
                              tobut@\(m\_j\^2\), {j, 
                                5}], 
									Table[
                              tobut@ν\_j, {j, 
                                5}], 
									Table[
                              tobut@Schiebe[i, "+"], {i, 
                                5}], 
	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Table[tobut@Schiebe[i, "-"], {i, 5}]}, 
							\(\({	
                          tobut /@ {PP, Δ\_\[Placeholder], 
                              u\_\[Placeholder], 𝒞\ }}\)\(
\)\)							], RowSpacings -> 0, 
\ \ \ \ \ \ ColumnSpacings -> 0]], 
                  NotebookDefault]}, 
	Editable -> True, 

              WindowToolbars -> {}, 
WindowSize -> {Fit, \ Fit}, 

              WindowMargins -> {{0, \ Automatic}, \ {0, Automatic}}, 

              WindowFrame -> "Palette", 
WindowElements -> {}, 

              WindowFrameElements -> "CloseBox", 

              WindowClickSelect -> False, 

              ScrollingOptions -> {"PagewiseScrolling" -> True}, 

              ShowCellBracket -> False, 

              CellMargins -> {{0, \ 0}, \ {0, \ 0}}, 
Active -> True, 

              CellOpen -> True, 
ShowCellLabel -> False, 

              ShowCellTags -> False, 

              ImageMargins -> {{0, \ 0}, \ {0, \ 0}}, 
Magnification -> 2, 
              Visible -> True]]; \)\(
\)\(SetOptions[pal1, 
          WindowMargins -> {{0, Automatic}, {Automatic, 
                0}}]; \)\(
\)\(SetOptions[pal1, Visible -> True, 
          WindowSize -> {Fit, Fit}]; \)\(
\)\(\ \)\(If[
          Head[pal2] === NotebookObject, 
	
          NotebookClose[pal2]]; \)\(
\)\(	\)\(pal2 = 
          NotebookPut[
            Notebook[{Cell[
                  BoxData[
                    GridBox[
							JJJ = 
                        Join[{tobut /@ {TFI[
                                  d + 2, {ν\_1, ν\_2, ν\_3, ν\_4, ν\_5}], TVI[d + 2, {ν\_1, ν\_2, ν\_3, ν\_4}], 
                                TJI[d + 2, {ν\_1, ν\_2, ν\_3}], 
                                TKI[d, {ν\_1, ν\_2, ν\_3}], 
                                TBI[d, {ν\_1, ν\_2}], 
                                TAI[
                                  d, {ν\_1}]}}, 
								\(\({tobut /@ {TFI[d, {ν\_1, ν\_2, ν\_3, ν\_4, ν\_5}], 
                                TVI[d, {ν\_1, ν\_2, ν\_3, ν\_4}], TJI[d, {ν\_1, ν\_2, ν\_3}], 
                                TKI[d - 2, {ν\_1, ν\_2, ν\_3}], 
                                TBI[d - 2, {ν\_1, ν\_2}], 
                                TAI[d - 
                                    2, {ν\_1}]}}\)\(
\)\)							], RowSpacings -> 0, 
\ \ \ \ \ \ ColumnSpacings -> 0]], 
                  NotebookDefault]}, 
	Editable -> True, 

              WindowToolbars -> {}, 
WindowSize -> {FitAll, \ FitAll}, 

              WindowMargins -> {{0, Automatic}, \ {0, Automatic}}, 

              WindowFrame -> "Palette", 
WindowElements -> {}, 

              WindowFrameElements -> "CloseBox", 

              WindowClickSelect -> False, 

              ScrollingOptions -> {"PagewiseScrolling" -> True}, 

              ShowCellBracket -> False, 

              CellMargins -> {{0, 0}, \ {0, 0}}, 
Active -> True, 

              CellOpen -> True, 
ShowCellLabel -> False, 

              ShowCellTags -> False, 

              ImageMargins -> {{0, 0}, \ {0, \ 0}}, 
Magnification -> 2, 
              Visible -> True]]; \)\(
\)\(SetOptions[pal2, 
          WindowMargins -> {{Automatic, 15}, {Automatic, 
                4}}]; \)\(
\)\(SetOptions[pal2, Visible -> True, 
          WindowSize -> {Fit, Fit}]; \)\(
\)\(	\)\(SetOptions[pal2, 
          WindowSize -> {Fit, Fit}]\)\(
\)\)	]; \)

OperatorApply

\(OperatorApplyF[x_ == y_] := 
      Block[{sch}, 
		sch\  = \ {Schiebe[i_, "+"]\^ni_. \ TFI[
                  d_, {n15__}] :> 
					TFI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] + ni, 
                  i]], 
				\(\(Schiebe[i_, "-"]\^ni_. \ TFI[
                  d_, {n15__}] :> 
					TFI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] - ni, 
                  i]]\)\(
\)\)			}; 
        Map[Collect[Expand[#] //. sch, TFI[__], Factor] &, 
			x == 
            y]]; \)

\(OperatorApplyV[x_ == y_] := 
      Block[{sch}, 
		sch\  = \ {Schiebe[i_, "+"]\^ni_. \ TVI[
                  d_, {n15__}] :> 
					TVI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] + ni, 
                  i]], 
				\(\(Schiebe[i_, "-"]\^ni_. \ TVI[
                  d_, {n15__}] :> 
					TVI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] - ni, 
                  i]]\)\(
\)\)			}; 
        Map[Collect[Expand[#] //. sch, TVI[__], Factor] &, 
			x == 
            y]]; \)

\(OperatorApplyJ[x_ == y_] := 
      Block[{sch}, 
		sch\  = \ {Schiebe[i_, "+"]\^ni_. \ TJI[
                  d_, {n15__}] :> 
					TJI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] + ni, 
                  i]], 
				\(\(Schiebe[i_, "-"]\^ni_. \ TJI[
                  d_, {n15__}] :> 
					TJI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] - ni, 
                  i]]\)\(
\)\)			}; 
        Map[Collect[Expand[#] //. sch, TJI[__], Factor] &, 
			x == 
            y]]; \)

\(OperatorApplyK[x_ == y_] := 
      Block[{sch}, 
		sch\  = \ {Schiebe[i_, "+"]\^ni_. \ TKI[
                  d_, {n15__}] :> 
					TKI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] + ni, 
                  i]], 
				\(\(Schiebe[i_, "-"]\^ni_. \ TKI[
                  d_, {n15__}] :> 
					TKI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] - ni, 
                  i]]\)\(
\)\)			}; 
        Map[Collect[Expand[#] //. sch, TKI[__], Factor] &, 
			x == 
            y]]; \)

\(OperatorApplyK[x_ == y_] := 
      Block[{sch}, 
		sch\  = \ {Schiebe[i_, "+"]\^ni_. \ TKI[
                  d_, {n15__}] :> 
					TKI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] + ni, 
                  i]], 
				\(\(Schiebe[i_, "-"]\^ni_. \ TKI[
                  d_, {n15__}] :> 
					TKI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] - ni, 
                  i]]\)\(
\)\)			}; 
        Map[Collect[Expand[#] //. sch, TKI[__], Factor] &, 
			x == 
            y]]; \)

\(OperatorApplyB[x_ == y_] := 
      Block[{sch}, 
		sch\  = \ {Schiebe[i_, "+"]\^ni_. \ TBI[
                  d_, {n15__}] :> 
					TBI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] + ni, 
                  i]], 
				\(\(Schiebe[i_, "-"]\^ni_. \ TBI[
                  d_, {n15__}] :> 
					TBI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] - ni, 
                  i]]\)\(
\)\)			}; 
        Map[Collect[Expand[#] //. sch, TBI[__], Factor] &, 
			x == 
            y]]; \)

\(OperatorApplyA[x_ == y_] := 
      Block[{sch}, 
		sch\  = \ {Schiebe[i_, "+"]\^ni_. \ TAI[
                  d_, {n15__}] :> 
					TAI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] + ni, 
                  i]], 
				\(\(Schiebe[i_, "-"]\^ni_. \ TAI[
                  d_, {n15__}] :> 
					TAI[d, 
                ReplacePart[{n15}, {n15}[\([\)\(i\)\(]\)] - ni, 
                  i]]\)\(
\)\)			}; 
        Map[Collect[Expand[#] //. sch, TAI[__], Factor] &, 
			x == 
            y]]; \)

MassDerivative

\(MassDerivative[
          STLI[any__, {
\ \ \ \ \ \ \ {al_, m1_}, {be_, m2_}, {ga_, 
                m3_}, {de_, m4_}, {ep_, m5_}}], {{0, d1_}, {0, d2_}, \ {0, 
              d3_}, \ {0, d4_}, \ {0, d5_}}]\) :=
\(\((\ \(Pochhammer[al, d1]\ Pochhammer[be, d2]\ Pochhammer[ga, 
                  d3]\)*
\ \ Pochhammer[de, d4]\ Pochhammer[ep, 
                d5]\ )\)\ *
\ 
          STLI[any, {{al + d1, m1}, \ {be + d2, m2}, \ {ga + d3, 
                m3}, \ {de + d4, m4}, \ {ep + d5, m5}}]\);

TarasovT

TarasovT[abrs__, qq_ /; Head[qq] =!= List] := 
  TarasovT[abrs, qq, {"g", "g", "g", "g", "g"}]

\(TarasovT[r_, \ s_, \ 
        qq_, \ {alp1_, alp2_, \ alp3_, alp4_, alp5_}]\  := 

      Block[{\ al1, al2, al3, al4, al5, alrul, new, Q1, Q11, Q12, Q2, Q22, 
          dum1, dum2}, 
\ \(If[alp1\  === \ 0, \ al1\  = \ 0]; \ 
          If[alp2\  === \ 0, \ al2\  = \ 0]; \)
\ \(If[alp3\  === \ 0, \ 
            al3\  = \ 0]; \)
\(If[alp4\  === \ 0, \ al4\  = \ 0]; \ 
          If[alp5\  === \ 0, \ al5\  = \ 0]; \)
\ \(alrul\  = \ 
            Select[{al1 :> I\ ParD[1], \ al2 :> I\ ParD[2], \ 
                al3 :> I\ ParD[3], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ al4 :> 
                  I\ ParD[4], \ 
                al5 :> I\ ParD[5]}, \ \(! MatchQ[#, \ 
                    0 :> _]\) &]; \)
\ \(Q1\  = \ 
            al3\ al5\  + \ al4\ al5\  + \ al2\ al3\  + \ 
              al3\ al4; \)
\ \(Q2\  = \ 
            al4\ al5\  + \ al3\ al5\  + \ al1\ al4\  + \ 
              al3\ al4; \)
\ \(Q11\  = 
            1/\((\(-4\))\)\ \((\ 
                al2\  + \ al4\  + \ al5\ )\); \)
\ \(Q22\  = 
            1/\((\(-4\))\)\ \((\ 
                al1\  + \ al3\  + \ al5\ )\); \)
\ \(Q12\  = 
            1/\((\(-2\))\)\ al5; \)
		\(new = 
\ 
            I^\((\(-r\) - s)\) \((\ 
                  D[\ Exp[\ 
                      I\ qq\ \((
\ \ \ \ be1\ Q1\  + \ be2\ Q2\  + 
                            be1^2\ Q11\  + be2^2\ Q22\  + \ 
                            be1\ be2\ Q12
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ )\)\ rho\ ], {be1, r}, {be2, 
                      s}
				]\ )\) /. 
\ \((\ 
                Join[alrul, \ {be1\  :> \ 0, be2\  :> \ 0, \ 
                    rho\  :> \ \(-Pi^2\)/
                          Pi^2\ DPlus\ }])\); \(\(Map[\((Select[#, \((FreeQ[#, ParD] && FreeQ[#, DPlus])\) &]\ TpD[qq, 
                      Select[#, \((\(! \((FreeQ[#, ParD] && 
                                  FreeQ[#, 
                                    DPlus])\)\))\) &]])\) &, \(\((Expand[new, 
                    ParD] + \ dum1\ dum2)\)\(
\)\)		] /. {dum1 :> 0, 
              dum2 :> 0}\)\(
\)\)\)		]; \)

\(\(\(
\)\(TarasovT[r_, \ s_, \ pp_, \ dp_ /; Head[dp] =!= List]\  := \ 
      TarasovT[r, s, pp, 
        dp, {"g", "g", "g", "g", "g"}]\)\)\)

(((Δ (Subscript[q, 1])))^a)(((Δ (Subscript[q, 2])))^b)(((p (Subscript[q, 1])))^r)(((p (Subscript[q, 2])))^s)

\(\(\(
\)\(TarasovT[a_, b_, r_, s_, pp_, 
        dp_, {alp1_, alp2_, alp3_, alp4_, alp5_}] := 
      Block[{al1, al2, al3, al4, al5, alrul, new, Q1, Q11, Q12, Q2, Q22, 
          dum1, dum2}, If[alp1 === 0, al1 = 0]; If[alp2 === 0, al2 = 0]; 
        If[alp3 === 0, al3 = 0]; If[alp4 === 0, al4 = 0]; 
        If[alp5 === 0, al5 = 0]; 
        alrul = Select[{al1 :> I\ ParD[1], 
              al2 :> I\ ParD[2], al3 :> I\ ParD[3], 
              al4 :> I\ ParD[4], 
              al5 :> 
                I\ ParD[5]}, \(\(\)\(! \((MatchQ[#1, 
                    0 :> _])\)\)\) &]; 
        Q1 = al3\ al5 + al4\ al5 + al2\ al3 + al3\ al4; 
        Q2 = al4\ al5 + al3\ al5 + al1\ al4 + al3\ al4; 
        Q11 = \(-\(1\/4\)\)\ \((al2 + al4 + al5)\); 
        Q22 = \(-\(1\/4\)\)\ \((al1 + al3 + al5)\); Q12 = \(-\(al5\/2\)\); 
        new = I\^\(\(-r\) - s - a - b\)\ ∂\_\({be1, r}, {be2, s}, {ga1, a}, {ga2, b}\)Exp[
                  I\ \((\((be1\ pp + ga1\ dp)\)\ Q1 + \((be2\ pp + 
                              ga2\ dp)\)\ Q2 + 
                        be1\ \((be1\ pp + 2\ ga1\ dp)\)\ Q11 + 
                        be2\ \((be2\ pp + 
                              2\ ga2\ dp)\)\ Q22 + \((be1\ be2\ pp + \((be1\ ga2 + be2\ ga1)\)\ dp)\)\ Q12)\)\ rho] /. 
            Join[alrul, {be1 :> 0, be2 :> 0, 
                ga1 :> 0, ga2 :> 0, 
                rho :> \(-\(\(π\^2\ DPlus\)\/π\^2\)\)}]; \((Select[#1, FreeQ[#1, ParD] && FreeQ[#1, DPlus] &]\ TpD[pp, 
                    Select[#1, \(\(\)\(! \((FreeQ[#1, ParD] && 
                              FreeQ[#1, DPlus])\)\)\) &]] &)\) /@ \((Expand[
                  new] + dum1\ dum2)\) /. {dum1 :> 0, 
            dum2 :> 0}]; \)\)\)

ApplyTarasovT

\(ApplyTarasovT[tij_, \ exp_] := 
      Block[{dump}, 
		dump = 
          Product[ParD[j]^dummy, {j, 5}]; 
		Expand[
            tij\ exp] /. 
\((STLI[de_, pp_, pe__List]\ TpD[tpp_, 
                  DPlus^i_. \ t_])\) :> 
	
            MassDerivative[
              STLI[\ de /. de :> \((de\  + \ 2\ i)\), pp, 
                pe], 
							\(\(t\ dump\  /. Times -> List\) /. 
                  Power[_, n_] :> {0, n}\) /. dummy -> 0]]\)

Misc

\(Prefactor1 /: 		Prefactor1[a_]^n_ := Prefactor1[a^n]\)

Colors

TFI TVI TFR TVR MakeFun Cayley CayleyD Cayleyu IFF ToArgs Block

MakeBoxes[Block, StandardForm] := 
      StyleBox["Block", FontSize -> 16, 
        FontWeight -> "Bold", 
        FontColor -> RGBColor[1, 0.2, 0.2]];

MakeBoxes[Cayley, StandardForm] := 
      StyleBox["Cayley", 
        FontColor -> RGBColor[1, 0, 0.4]];

MakeBoxes[CayleyD, StandardForm] := 
      StyleBox["CayleyD", 
        FontColor -> RGBColor[1, 0.2, 1]];

MakeBoxes[Cayleyu, StandardForm] := 
      StyleBox["Cayleyu", 
        FontColor -> RGBColor[0.8, 0.8, 0.4]];

MakeBoxes[IFF, StandardForm] := 
      StyleBox["IFF", 
        FontColor -> RGBColor[0.6, 0.4, 1]];

MakeBoxes[MakeFun, StandardForm] := 
      StyleBox["MakeFun", 
        FontColor -> RGBColor[1, 0.6, 0.6]];

MakeBoxes[TFI, StandardForm] := 
      StyleBox["TFI", FontColor -> RGBColor[1, 0.6, 0]];

MakeBoxes[TFR, StandardForm] := 
      StyleBox["TFR", FontColor -> RGBColor[0, 0.6, 0]];

MakeBoxes[ToArgs, StandardForm] := 
      StyleBox["ToArgs", 
        FontColor -> RGBColor[0.2, 0.6, 1]];

MakeBoxes[TVI, StandardForm] := 
      StyleBox["TVI", 
        FontColor -> RGBColor[0.2, 0.4, 0.8]];

MakeBoxes[TVR, StandardForm] := 
      StyleBox["TVR", FontColor -> RGBColor[0.2, 0.8, 1]]

PQ

PQ[_Integer?Positive]  :=  True;

PNQ

PNQ[0] = True;

PNQ[_Integer?Positive]  :=  True;

FactorC

FactorC[z_] := FactorC[z] = Factor[z]

TFI[depp__,{a___,b_Integer,c___}]:=TFI[depp,{a,{b,0},c}];
TFI[depp__,{a___,{0,m_/;m=!=0},b___}]:=TFI[depp,{a,{0,0},b}];
TFI[a__,{0,0},b__List]:=TFI[a,b];
TFI[d_,pp_,dp_,{0,0},{x1_,x2_,x3_,x4_,x5_},list_List]:=
    TFI[d,pp,{x1,x2,x3,x4,x5},list];
TFI[d_,pp_,dp_,{a_,b_},{0,0,0,0,0},list_List]:=TFI[d,pp,dp,{a,b},list];
TFI[d_,pp_,dp_/;Head[dp]=!=List,{0,0,0,0,0},list_List]:=TFI[d,pp,list];
TFI[d_,pp_,dp_/;Head[dp]=!=List,list_List]:=TFI[d,pp,list];
TFI[d_,pp_,dp_/;Head[dp]=!=List,{x1_,x2_,x3_,x4_,x5_},list_List]:=
    TFI[d,pp,{x1,x2,x3,x4,x5},list];
TFI[__,{{_,_},{0,0},{_,_},{0,0},{_,0}}]:=0;
TFI[__,{{0,0},{_,_},{0,0},{_,_},{_,0}}]:=0;
TFI[__,{{0,0},{_,_},{_,0},{_,_},{0,0}}]:=0;
TFI[__,{{_,0},{_,_},{0,0},{_,_},{0,0}}]:=0;
TFI[__,{{_,_},{_,0},{_,_},{0,0},{0,0}}]:=0;
TFI[__,{{_,_},{0,0},{_,_},{_,0},{0,0}}]:=0;
TFI[__,{{0,0},{0,0},{0,0},{_,0},{_,_}}]:=0;
TFI[__,{{0,0},{0,0},{_,0},{0,0},{_,_}}]:=0;
TFI[__,{{0,0},{_,0},{0,0},{0,0},{_,_}}]:=0;
TFI[__,{{_,0},{0,0},{0,0},{0,0},{_,_}}]:=0;
TFI[__,{{_,_},{_,0},{0,0},{0,0},{0,0}}]:=0;
TFI[__,{{_,0},{_,_},{0,0},{0,0},{0,0}}]:=0;
TFI[__,{{_,_},{0,0},{0,0},{_,0},{0,0}}]:=0;
TFI[__,{{_,0},{0,0},{0,0},{_,_},{0,0}}]:=0;
TFI[__,{{0,0},{_,0},{_,_},{0,0},{0,0}}]:=0;
TFI[__,{{0,0},{_,_},{_,0},{0,0},{0,0}}]:=0;
TFI[__,{{0,0},{0,0},{_,_},{_,0},{0,0}}]:=0;
TFI[__,{{0,0},{0,0},{_,0},{_,_},{0,0}}]:=0;

If[$Notebooks,

mbt[z_] := ToBoxes[z, TraditionalForm];
  redblue[z_ /; Head[z] =!= Plus] := mbt[z];
  redblue[(z_) - 1] := StyleBox[mbt[z], FontColor -> RGBColor[1, 0, 0]];
  redblue[(z_Subscript) - 2] :=
   StyleBox[SubscriptBox[OverscriptBox[z[[1]], "_"], z[[2]]],
    FontColor -> RGBColor[1, 0, 0]];
  redblue[(z_) - 2] :=
   StyleBox[OverscriptBox[mbt[z], "_"], FontColor -> RGBColor[1, 0, 0]];
  redblue[(z_) - 3] :=
   StyleBox[UnderscriptBox[OverscriptBox[mbt[z], "_"], "_"],
    FontColor -> RGBColor[1, 0, 0]];
  redblue[(z_) + 1] := StyleBox[mbt[z], FontColor -> RGBColor[0, 0, 1]];
  redblue[(z_) + 2] :=
   StyleBox[OverscriptBox[mbt[z], "_"], FontColor -> RGBColor[0, 0, 1]];
  redblue[(z_Subscript) + 2] :=
   StyleBox[SubscriptBox[OverscriptBox[z[[1]], "_"], z[[2]]],
    FontColor -> RGBColor[0, 0, 1]];

  TFI/: MakeBoxes[TFI[d_, pp_, {ur__}, {den__}], TraditionalForm] :=
    InterpretationBox @@
      {SubsuperscriptBox[StyleBox["F", Rule[SingleLetterItalics, False],
         Rule[FontWeight, "Bold"]], RowBox @@ {redblue /@ {den}},
        RowBox[{"(", ToBoxes[d, TraditionalForm], ")", " ", ur}]],
       TFI[d, pp, {ur}, {den}], Editable -> True} /;
     MemberQ[{StandardForm, TraditionalForm}, TraditionalForm];

  TFI/: MakeBoxes[TFI[d_, pp_, dp_, any__, {den__}], TraditionalForm] :=
    (InterpretationBox @@
       {SubsuperscriptBox[StyleBox["F",
          Rule[SingleLetterItalics, False], Rule[FontWeight, "Bold"]],
         RowBox @@ {redblue /@ {den}},
         RowBox[{"(", ToBoxes[d, TraditionalForm], ")", " ",
           Sequence @@ Flatten[{any}]}]], TFI[d, pp, dp, any, {den}],
        Editable -> True}) /; Head[dp] =!= List;

TFI/:
   MakeBoxes[TFI[dpp__, {den__}], TraditionalForm] :=
    InterpretationBox @@
      {SubsuperscriptBox[StyleBox["F", Rule[SingleLetterItalics, False],
         Rule[FontWeight, "Bold"]], RowBox @@ {redblue /@ {den}},
        RowBox[{"(", ToBoxes[First[{dpp}], TraditionalForm], ")"}]],
       TFI[dpp, {den}], Editable -> True} /;
     MemberQ[{StandardForm, TraditionalForm}, TraditionalForm] &&
      Head[Last[{dpp}]] =!= List;

];

TJI

TJI[de_, 
        pp_, {{n1_Integer, m1_}, {n2_Integer, m2_}, {n3_Integer, 
            m3_}}] := (TJI[de, pp, {{n1, m1}, {n2, m2}, {n3, m3}}] = 
            TJI[de, pp, 
              Reverse[Sort[{{n1, m1}, {n2, m2}, {n3, 
                      m3}}]]]) /; ! (OrderedQ[{{n3, m3}, {n2, m2}, {n1, m1}}])

TJI[d, pp, {{1, m1}, {1, m2}, {1, m3}}]

TJI[d_, 0, {{_, 0}, {_, 0}, {_, 0}}] := 0

TBI

TBI[de_, 
        pp_, {{n1_Integer, m1_}, {n2_Integer, m2_}}]:=(TJI[de, pp, {{n1, m1}, {n2, m2}}]=TBI[de, pp, 
              Reverse@Sort@{{n1, m1}, {n2, m2}}])/;! (OrderedQ[{{n2, m2}, {n1, 
                  m1}}])

TBI[d_, 0, {{_, 0}, {_, 0}}] := 0

TAI

TAI[de_, 
          pp_, {{n1_Integer, 0}}]:=0;

Definitions and notation

Δ𝒞=-(1/2)Det[ (({{0, 1, 1, 1, 1}, {1, 0, (Subscript[μ, 6]), (Subscript[μ, 4]), (Subscript[μ, 3])}, {1, (Subscript[μ, 6]), 0, (Subscript[μ, 2]), (Subscript[μ, 1])}, {1, (Subscript[μ, 4]), (Subscript[μ, 2]), 0, (Subscript[μ, 5])}, {1, (Subscript[μ, 3]), (Subscript[μ, 1]), (Subscript[μ, 5]), 0}}))];

\(checkli\  = \ {
		𝒞 -> Δ𝒞, 
Δ\_i_ :> ∂\_\(μ\_i\)Δ𝒞, 
		μ\_6 :> PP, μ\_i_ :> 
          ToExpression["m" <> ToString[i]]^2, 
		m\_1 :> \@m1\^2, 
        m\_2 :> \@m2\^2, m\_3 :> \@m3\^2, m\_4 :> \@m4\^2, 
        m\_5 :> \@m5\^2, 
Δ\_\(i_, j_, k_\) :> μ\_i\%2 + μ\_j\%2 + μ\_k\%2 - 
            2 \((\(μ\_i\) μ\_j + \(μ\_j\) μ\_k + \(μ\_k\) μ\_i)\), 

        u\_\(i_, j_, k_\) :> μ\_i - μ\_j - μ\_k, 
Σ\_i_ :> 
          3\ d\  - 2\ \((∑\_\(j = 1\)\%i ν\_j)\) - 
            2, 
	ν\_1 :> n1, ν\_2 :> 
          n2, ν\_3 :> n3, ν\_4 :> 
          n4, ν\_5 :> n5}; \)

Define symmetry property for (Subscript[Δ, i, j, k])

(Subscript[Δ, i_, j_, k_]) := (Subscript[Δ, Sequence @@ Sort[{i, j, k}]]) /; ! OrderedQ[{i, j, k}]

(Subscript[Δ, 3, 1, 2])

ToArgs

ToArgs[z_]:=Expand[z //. checkli];

ToArgs[(Subscript[Σ, 4])] // InputForm

Cayley

This is (40).  ( p = (Sqrt[(p^2)]))

setd[hCayley[(Subscript[m, 1]):_,(Subscript[m, 2]):_,(Subscript[m, 3]):_,(Subscript[m, 4]):_,(Subscript[m, 5]):_,p_],set[hCayley[(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),p],factor[ToArgs[𝒞]/.(p^2)->(p^2)]]]/.{hCayley:>Cayley,setd :> SetDelayed,set :> Set,factor :> Factor}

Cayley[i]=(Subscript[Δ, i])

Do[setd[hCayley[i][{(Subscript[m, 1]):_,(Subscript[m, 2]):_,(Subscript[m, 3]):_,(Subscript[m, 4]):_,(Subscript[m, 5]):_,p_}],set[hCayley[i][{(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),p}],factor[ToArgs[(Subscript[Δ, i])]/.(p^2)->(p^2)]]]/.{hCayley:>Cayley,setd :> SetDelayed,set :> Set,factor :> Factor},{i, 6}]

ToArgs[(Subscript[Δ, 1, 2, 6])]

Cayley[i_, j_, k_][m_List]:=Cayley[i, j, k][m]=\(Block[{mm = m}, 
			While[Length[mm] < 7, 
              AppendTo[mm, Last[mm]]]; 
			FactorC[
              mm[[i]]\^4 - 
                2\ mm[[j]]\^2\ mm[[i]]\^2 - 
                2\ mm[[k]]\^2\ mm[[i]]\^2 + 
                mm[[j]]\^4 + 
                mm[[k]]\^4 - 
                2\ mm[[j]]\^2\ mm[[k]]\^2]]\);

Cayleyu[i_, j_, k_][m_List]:=Cayleyu[i, j, k][mm]=\(Block[{mm = m}, 
					While[Length[mm] < 7, 
              AppendTo[mm, Last[mm]]]; 
            FactorC[mm[[i]]\^2 - 
                mm[[j]]\^2 - 
                mm[[k]]\^2]]\);

CayleyD[i]=((Subscript[Δ, i])/Δ)

Do[setd[hCayleyD[i][{(Subscript[m, 1]):_,(Subscript[m, 2]):_,(Subscript[m, 3]):_,(Subscript[m, 4]):_,(Subscript[m, 5]):_,p_}],set[hCayleyD[i][{(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),p}],factor[((ToArgs[(Subscript[Δ, i])]/.(p^2)->(p^2))/(𝒞[(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),p]))]]]/.{hCayleyD:>CayleyD,setd :> SetDelayed,set :> Set,factor :> Factor,𝒞:>Cayley},{i, 6}]

setd[hCayleyD[1, 2, 5][{(Subscript[m, 1]):_,(Subscript[m, 2]):_,(Subscript[m, 3]):_,(Subscript[m, 4]):_,(Subscript[m, 5]):_,p_}],set[hCayleyD[1, 2, 5][{(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),p}],factor[((ToArgs[(Subscript[Δ, 1, 2, 5])]/.(p^2)->(p^2))/(𝒞[(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),p]))]]]/.{hCayleyD:>CayleyD,setd :> SetDelayed,set :> Set,factor :> Factor,𝒞:>Cayley}

setd[hCayleyD[1, 3, 6][{(Subscript[m, 1]):_,(Subscript[m, 2]):_,(Subscript[m, 3]):_,(Subscript[m, 4]):_,(Subscript[m, 5]):_,p_}],set[hCayleyD[1, 3, 6][{(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),p}],factor[((ToArgs[(Subscript[Δ, 1, 3, 6])]/.(p^2)->(p^2))/(𝒞[(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),p]))]]]/.{hCayleyD:>CayleyD,setd :> SetDelayed,set :> Set,factor :> Factor,𝒞:>Cayley}

setd[hCayleyD[3, 4, 5][{(Subscript[m, 1]):_,(Subscript[m, 2]):_,(Subscript[m, 3]):_,(Subscript[m, 4]):_,(Subscript[m, 5]):_,p_}],set[hCayleyD[3, 4, 5][{(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),p}],factor[((ToArgs[(Subscript[Δ, 3, 4, 5])]/.(p^2)->(p^2))/(𝒞[(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),p]))]]]/.{hCayleyD:>CayleyD,setd :> SetDelayed,set :> Set,factor :> Factor,𝒞:>Cayley}

ToArgs[(Subscript[Δ, i, j, k])]

ToArgs[(Subscript[Δ, 1, 2, 6])]

ToArgs[(Subscript[u, i, j, k])]

Utility Functions

MakeFun and friends

This function generates the recursion function definitions for 
TARCER starting from the entered relations.

The idea is to recurse TFR into TFI's step by step.

Define: S cD[ijk__] := \(TraditionalForm\`\(𝒞\^\(-1\)\) Δ\_ijk\)

\(nutonandΔu[w_, 
        5] := \(w /. {ν\_1 :> n1, ν\_2 :> 
              n2, ν\_3 :> n3, ν\_4 :> 
              n4, ν\_5 :> 
              n5, Δ\_ijk__ :> 𝒞\ cD[
                  ijk], u\_ijk__ :> cu[ijk], 
		D\_ijk__ :> 
              TA[D, ijk], ρ\_ijk__ :> 
              TA[ρ, ijk], ϕ\_\(\(ijk__\)\(\ \)\) :> 
              TA[ϕ, ijk], 
			σ\_\(i_, j_, k_\) :> 
              TA[σ, d, i, j, k, ν\_i, ν\_j, ν\_k], 
            h\_\(i_, j_, k_\) :> 
              TA[h, d, i, j, k, ν\_i, ν\_j, ν\_k], 
            S\_\(i_, j_, k_\) :> 
              TA[S, d, i, j, 
                k, ν\_i, ν\_j, ν\_k], 
			\(\(Σ\_i_ :> ToArgs[Σ\_i]\)\(
\)\)		}\) /. {	ν\_1 :> n1, ν\_2 :> n2, ν\_3 :> 
            n3, ν\_4 :> n4, ν\_5 :> n5, 
          m\_1 :> m1, m\_2 :> m2, m\_3 :> m3, m\_4 :> m4, 
          m\_5 :> m5}; \)

nutonandΔu[
        w_] := w /. {(Subscript[ν, 1]) :> n1, (Subscript[ν, 2]) :> 
              n2, (Subscript[ν, 3]) :> n3, (Subscript[ν, 4]) :> 
              n4, (Subscript[ν, 5]) :> 
              n5, (Subscript[Δ, ijk])__ :> cD[ijk], 
            (Subscript[u, ijk])__ :> cu[ijk], 
            (Subscript[D, ijk])__ :> TA[D, ijk], (Subscript[ρ, ijk])__ :> 
              TA[ρ, ijk], (Subscript[ϕ, ijk])__ :> 
              TA[ϕ, ijk], (Subscript[σ, i_, j_, k_]) :> 
              TA[σ, d, i, j, k, (Subscript[ν, i]), (Subscript[ν, j]), (Subscript[ν, k])], 
            (Subscript[h, i_, j_, k_]) :> 
              TA[h, d, i, j, k, (Subscript[ν, i]), (Subscript[ν, j]), (Subscript[ν, k])], 
            (Subscript[S, i_, j_, k_]) :> 
              TA[S, d, i, j, 
                k, (Subscript[ν, i]), (Subscript[ν, j]), (Subscript[ν, k])], (Subscript[Σ, i])_ :> ToArgs[(Subscript[Σ, i])]} /. {(Subscript[ν, 1]) :> 
            n1, (Subscript[ν, 2]) :> n2, (Subscript[ν, 3]) :> 
            n3, (Subscript[ν, 4]) :> n4, (Subscript[ν, 5]) :> n5, 
          (Subscript[m, 1]) :> m1, (Subscript[m, 2]) :> m2, 
          (Subscript[m, 3]) :> m3, (Subscript[m, 4]) :> m4, 
          (Subscript[m, 5]) :> m5};

addPeP[TFR[d_, pp_, li1_, rest_]] := TFR[d, pp, li1, rest];

addPeP[TFI[d_, pp_, li1_List]] := TFI[d, pp, li1];

addPeP[TFR[d_, pp_, dp_Symbol, li1_, rest_]] := 
      TFR[d, pp, dp, li1, rest];

addPeP[TFI[d_, pp_, dp_Symbol, li1_List]] := 
      TFI[d, pp, dp, li1];

addPeP[TVI[d_, pp_, li1_List]] := TVI[d, pp, li1];

nnnnnaddPeP[any_[d_, pp_, li1_List, rest__List]] := 
      any[d, pp, li1, rest] /; any =!= List

addPeP[z_List] := Append[z, (Sqrt[PP])];

addPeP[TFI[d_, z_List]] := TFI[d, PP, z];

addPeP[TVI[d_, z_List]] := TVI[d, PP, z];

addPeP[TJI[d_, z_List]] := TJI[d, PP, z];

addPeP[TFR[d_, z_List]] := TFR[d, PP_, z];

addPeP[TFR[d_, pl_List, z_List]] := TFR[d, PP_, pl, z];

addPeP[TVR[d_, z__List]] := TVR[d, PP_, z];

addPeP[TJR[d_, z__List]] := TJR[d, PP_, z];

nutomass[
        w_] := w /. {(Subscript[ν, 1]) + i1_.  :> {n1 + i1, 
                  m1}, (Subscript[ν, 2]) + i2_.  :> {n2 + i2, 
                  m2}, (Subscript[ν, 3]) + i3_.  :> {n3 + i3, 
                  m3}, (Subscript[ν, 4]) + i4_.  :> {n4 + i4, 
                  m4}, (Subscript[ν, 5]) + i5_.  :> {n5 + i5, 
                  m5}} /. {in1_Integer, in2_Integer, in3_Integer, 
              in4_Integer} :> {{in1, m1}, {in2, m2}, {in3, 
                m3}, {in4, m4}} /. {in1_Integer, in2_Integer, 
            in3_Integer} :> {{in1, m1}, {in2, m2}, {in3, 
              m3}};

Clear[maybeF];

maybeF[z_Times] := 
      FactorC[z] /; 
        FreeQ[z, TFI] && FreeQ[z, TVI] && FreeQ[z, TJI] && 
          Length[Cases[z, cD[__], -1]] > 1;

maybeF[z_] := z; SetAttributes[top, Listable];

This is the function for the information of side conditions.

SetAttributes[IFF,HoldAll];

Clear[MakeFun];

MakeFun[c_. TFI[t__]==rhs_,eqn_String,IFF[conds_],mrel_ : {}]:=Block[{nm},nmrel = Select[Flatten[{mrel}], FreeQ[#1, Factor] &];(HoldForm@@{setdel[addPeP[TFR@@top /@ nutomass[{t}]]/.nmrel,Hold[Condition][Hold[CompoundExpression][	Hold[TComment][
                                        eqn, addPeP[nutomass[TFR[t]]] /. 
                                        TFR -> TFIC /. {Pattern :> pat, 
                                        Blank :> blank} /. pat[PP, 
                                        blank[]] :> PP /. nmrel],Collect[TT=Expand3[nutonandΔu[(1/c)(rhs/.TFI[te__]:>addPeP[nutomass[TFI[te]]]),5]],TFI[__],maybeF[(Collect[
                                        Numerator[#1], cD[__], 
                                        Factor]/Collect[Denominator[#1], 
                                        cD[__], Factor] &) /@ 
                                        Factor[#1]] &]/.nmrel],nutonandΔu[IFF[conds],5]]]/.top[i_Integer] -> i/.top :> (patternTest[pattern[#1, blank[]], PQ] &)/.patternTest[pattern[dm_/;MemberQ[{d,(p^2),DP,(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5])},dm],blank[]],_]:>pattern[dm, 
                                        blank[]]/.
					{blank :> 
                                        Blank, 
                                        pattern :> Pattern, 
                                        patternTest :> 
                                        PatternTest}}/.cD[ijk__]->Hold[CayleyD[ijk]][addPeP[
                                        BLA = nutomass[Last[{t}]] /. 
                                        nmrel /. {nuu_, ma_} :> 
                                        ma]]/.addPeP -> Identity/.Hold[CompoundExpression] -> 
                                  CompoundExpression/.Hold[Condition][a_,IFF[b_]]->a /; b/.Hold[Set] -> Set/.Hold[TComment] -> TComment/.Hold[CayleyD[ijk__]]->CayleyD[ijk]/.{setdel :> SetDelayed})/.Identity -> identit/.identit[a_] -> a];

MakeFun[c_. TFI[t__]==rhs_,eqn_String,mrel_ : {}]:=Block[{nm},nmrel = 
                  Select[Flatten[{mrel}], FreeQ[#1, Factor] &];(HoldForm@@{setdel[addPeP[TFR@@top /@ nutomass[{t}]]/.nmrel,Hold[CompoundExpression][Hold[TComment][
                                        eqn, addPeP[nutomass[TFR[t]]] /. 
                                        TFR -> TFIC /. {Pattern :> pat, 
                                        Blank :> blank} /. pat[PP, 
                                        blank[]] :> PP /. nmrel],Collect[TT=Expand3[nutonandΔu[(1/c)(rhs/.TFI[te__]:>addPeP[nutomass[TFI[te]]])]],TFI[__],maybeF[(Collect[
                                        Numerator[#1], cD[__], 
                                        Factor]/Collect[Denominator[#1], 
                                        cD[__], Factor] &) /@ 
                                        Factor[#1]] &]/.nmrel]]/.top[i_Integer] -> i/.top :> (patternTest[pattern[#1, blank[]], PQ] &)/.patternTest[pattern[dm_/;MemberQ[{d,(p^2),DP,(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5])},dm],blank[]],_]:>pattern[
                                        dm, blank[]]/.	{blank :> Blank, 
                                        pattern :> Pattern, 
                                        patternTest :> 
                                        PatternTest}}/.cD[ijk__]->Hold[CayleyD[ijk]][addPeP[nutomass[
                                        Last[{t}]] /. addPeP -> Identity /. 
                                        nmrel /. {nuu_, ma_} :> 
                                        ma]]/.Hold[CompoundExpression] -> 
                                      CompoundExpression/.Hold[Condition][a_,IFF[b_]]->a /; b/.Hold[Set] -> Set/.Hold[TComment] -> TComment/.Hold[CayleyD[ijk__]]->CayleyD[ijk]/.{setdel :> SetDelayed})/.addPeP -> Identity/.Identity -> identit/.identit[a_] -> a]/;Head[mrel]=!=IFF;

MakeFun[c_. TVI[t__]==rhs_,eqn_String,IFF[conds_],mrel_ : {}]:=Block[{nm},nmrel = Select[Flatten[{mrel}], FreeQ[#1, Factor] &];nmrel=Join[nmrel,{(Subscript[m, 1]):>(Subscript[m, 1]),(Subscript[m, 2]):>(Subscript[m, 2]),(Subscript[m, 3]):>(Subscript[m, 3]),(Subscript[m, 4]):>(Subscript[m, 4]),(Subscript[Σ, i])_ :> 
                          ToArgs[(Subscript[Σ, i])]}];HoldForm@@{setdel[addPeP[TVR@@top /@ nutomass[{t}]]/.nmrel,Hold[Condition][Hold[CompoundExpression][Hold[TComment][
                                        eqn, addPeP[nutomass[TVR[t]]] /. 
                                        TVR -> TVIC /. {Pattern :> pat, 
                                        Blank :> blank} /. pat[PP, 
                                        blank[]] :> PP /. nmrel],Collect[TT=Expand3[nutonandΔu[(1/c)(rhs/.TVI[te__]:>addPeP[nutomass[TVI[te]]])]],TVI[__],maybeF[(Collect[
                                        Numerator[#1], cu[__], 
                                        Factor]/Collect[Denominator[#1], 
                                        cu[__], Factor] &) /@ 
                                        Factor[#1]] &]/.𝒞 -> 1/.nmrel],nutonandΔu[IFF[conds]]]]/.top[
                                        i_Integer] -> i/.top :> (patternTest[pattern[#1, blank[]], PQ] &)/.patternTest[pattern[dm_/;MemberQ[{d,(p^2),(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4])},dm],blank[]],_]:>pattern[
                                        dm, blank[]]/.	{blank :> Blank, 
                                        pattern :> Pattern, 
                                        patternTest :> 
                                        PatternTest}}/.
			HoldPattern[
                                        TA[aijk__]] -> Hold[
                                        TA[aijk]][
                                        addPeP[nutomass[Last[{t}]] /. 
                                        nmrel /. {nuu_, ma_} :> 
                                        ma]]/.
			cD[
                                        ijk__] -> Hold[Cayley[ijk]][
                                        addPeP[nutomass[Last[{t}]] /. 
                                        nmrel /. {nuu_, ma_} :> 
                                        ma]]/.cu[ijk__]->Hold[Cayleyu[ijk]][addPeP[nutomass[
                                        Last[{t}]] /. addPeP -> Identity /. 
                                        nmrel /. {nuu_, ma_} :> 
                                        ma]]/.Hold[CompoundExpression] -> 
                                      CompoundExpression/.Hold[Condition][a_,IFF[b_]]->a /; b/.Hold[Set] -> Set/.Hold[TComment] -> TComment/.addPeP -> Identity/.Hold[Cayleyu[ijk__]]->Cayleyu[ijk]/.Hold[Cayley[ijk__]]->Cayley[ijk]/.
			Hold[TA] -> TA/.{setdel :> SetDelayed}/.Identity -> identit/.identit[a_] -> a];

MakeFun[c_. TVI[t__]==rhs_,eqn_String,mrel_ : {}]:=Block[{nm},nmrel = 
                  Select[Flatten[{mrel}], FreeQ[#1, Factor] &];nmrel=Join[nmrel,{(Subscript[m, 1]):>(Subscript[m, 1]),(Subscript[m, 2]):>(Subscript[m, 2]),(Subscript[m, 3]):>(Subscript[m, 3]),(Subscript[m, 4]):>(Subscript[m, 4]),(Subscript[Σ, i])_:>ToArgs[(Subscript[Σ, i])]}];HoldForm@@{setdel[addPeP[TVR@@top /@ nutomass[{t}]]/.nmrel,Hold[CompoundExpression][Hold[TComment][
                                        eqn, addPeP[nutomass[TVR[t]]] /. 
                                        TVR -> TVIC /. {Pattern :> pat, 
                                        Blank :> blank} /. pat[PP, 
                                        blank[]] :> PP /. nmrel]
						,Collect[TT=Expand3[nutonandΔu[(1/c)(rhs/.TVI[te__]:>addPeP[nutomass[TVI[te]]])]],TVI[__],maybeF[(Collect[
                                        Numerator[#1], cu[__], 
                                        Factor]/Collect[Denominator[#1], 
                                        cu[__], Factor] &) /@ 
                                        Factor[#1]] &]/.nmrel/.𝒞 -> 
                                        1]]/.top[i_Integer] -> i/.top :> (patternTest[pattern[#1, blank[]], PQ] &)/.patternTest[pattern[dm_/;MemberQ[{d,(p^2),(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4])},dm],blank[]],_]:>pattern[
                                        dm, blank[]]/.{blank :> Blank, 
                                        pattern :> Pattern, 
                                        patternTest :> 
                                        PatternTest}}/.cD[ijk__]->Hold[Cayley[ijk]][addPeP[nutomass[Last[{t}]] /. 
                                        nmrel /. {nuu_, 
                                        ma_Symbol} :> ma]]/.cu[ijk__]->Hold[Cayleyu[ijk]][addPeP[nutomass[Last[{t}]] /. 
                                        nmrel /. {nuu_, 
                                        ma_Symbol} :> ma]]/.Hold[CompoundExpression] -> 
                                    CompoundExpression/.Hold[Set] -> Set/.Hold[TComment] -> TComment/.addPeP -> Identity/.Hold[Cayleyu[ijk__]]->Cayleyu[ijk]/.Hold[Cayley[ijk__]]->Cayley[ijk]/.{setdel :> SetDelayed}/.Identity -> identit/.identit[a_] -> a]/;Head[mrel]=!=IFF;

MakeFun[c_.  TJI[t__] == rhs_,eqn_String,IFF[conds_],mrel_ : {}]:=Block[{nm, nmrel,  result},𝒞 = 1;nmrel = Select[Flatten[{mrel}], FreeQ[#1, Factor] &];nmrel=Join[nmrel,{(Subscript[m, 1]):>(Subscript[m, 1]),(Subscript[m, 2]):>(Subscript[m, 2]),(Subscript[m, 3]):>(Subscript[m, 3]),(Subscript[m, 4]):>(Subscript[m, 4]),(Subscript[Σ, i])_:>ToArgs[(Subscript[Σ, i])]}];
		result = HoldForm@@{setdel[addPeP[TJR @@ top /@ 
                                        nutomass[{tT = t}]] /. nmrel,Hold[Condition][\(\(Hold[
                                        CompoundExpression]\)[\(Hold[
                                        TComment]\)[eqn, \(\(\(addPeP[
                                        nutomass[TJR[t]]] /. TJR -> 
                                        TJIC\) /. {Pattern :> pat, Blank :> 
                                        blank}\) /. pat[PP, blank[]] :> PP\) /. 
                                        nmrel]	, TJ = \(Collect[TT = Expand3[
                                        nutonandΔu[\(1\/c\) \((rhs /. TJI[te__] :> addPeP[nutomass[TJI[te]]])\)]], TJI[__], 
                                        maybeF[\((Collect[Numerator[#1], 
                                        cu[__], 
                                        Factor]/Collect[Denominator[#1], 
                                        cu[__], Factor] &)\) /@ 
                                        Factor[#1]] &] /. 𝒞 -> 1\) /. nmrel]\),nutonandΔu[IFF[conds]]]]/.top[
                                        i_Integer] -> i/.top :> (patternTest[pattern[#1, blank[]], PQ] &)/.patternTest[pattern[dm_/;MemberQ[{d,(p^2),(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4])},dm],blank[]],_]:>pattern[
                                        dm, blank[]]/.{blank :> Blank, 
                                        pattern :> Pattern, 
                                        patternTest :> 
                                        PatternTest}}/.HoldPattern[
                                        TA[aijk__]] -> Hold[TA][
                                        aijk][
                                        addPeP[nutomass[Last[{t}]] /. 
                                        nmrel /. {nuu_, ma_} :> 
                                        ma]]/.cD[ijk__]->Hold[Cayley[ijk]][addPeP[nutomass[Last[{t}]] /. 
                                        nmrel /. {nuu_, 
                                        ma_Symbol} :> ma]]/.cu[ijk__]->Hold[Cayleyu[ijk]][addPeP[nutomass[Last[{t}]] /. 
                                        nmrel /. {nuu_, 
                                        ma_Symbol} :> ma]]/.Hold[CompoundExpression] -> 
                                  CompoundExpression/.Hold[Condition][a_,IFF[b_]]->a /; b/.Hold[Set] -> Set/.Hold[TComment] -> TComment/.Hold[Cayleyu[ijk__]]->Cayleyu[ijk]/.Hold[Cayley[ijk__]]->Cayley[ijk]/.Hold[TA] -> TA/.{setdel :> SetDelayed};𝒞 =  . ; result];

SetAttributes[paTT, HoldFirst];

MakeFun[c_. TKI[t__]==rhs_,eqn_String,IFF[conds_],mrel_ : {}]:= MakeFun[c TJI[t] == rhs /. TKI -> TJI, eqn, IFF[conds], 
                mrel] /. {PP :> 0, Sqrt[PP] :> 0, Pattern :> paTT} /. 
            paTT[0, _] -> 0 /. paTT -> Pattern /. 
        TJIC[d_, li_List] :> TJIC[d, 0, li]

MakeFun[c_. TBI[t__]==rhs_,eqn_String,IFF[conds_],mrel_ : {}]:= \(MakeFun[c\ TJI[t] == rhs /. TBI -> TJI, eqn, IFF[conds], 
            mrel] /. {
			TJI :> TBI, TJR :> TBR, TJIC :> TBIC}\);

MakeFun[c_. TAI[t__]==rhs_,eqn_String,IFF[conds_],mrel_ : {}]:= \(MakeFun[c\ TJI[t] == rhs /. TAI -> TJI, eqn, IFF[conds], 
            mrel] /. {
			TJI :> TAI, TJR :> TAR, TJIC :> TAIC}\);

Interchange

Interchange[z_, i_ ↔ j_]:=\(νexplicit[
            z] /. {ν\_i :> ν\_j, ν\_j :> ν\_i, m\_i :> m\_j, 
            m\_j :> m\_i, Δ\_ij__ :> \((Δ\_ij /. {i :> j, j :> i})\), 
            u\_ij__ :> \((u\_ij /. {i :> j, 
                    j :> i})\), 
		h\_ij__ :> \((h\_ij /. {i :> j, 
                    j :> i})\), ϕ\_ij__ :> \((ϕ\_ij /. {i :> j, 
                    j :> i})\), ρ\_ij__ :> \((ρ\_ij /. {i :> j, 
                    j :> i})\), σ\_ij__ :> \((σ\_ij /. {i :> j, 
                    j :> i})\), D\_ij__ :> \((D\_ij /. {i :> j, j :> i})\)}\)/.{TFI[depp__, args_List]:>TFI[depp, 
                  Sort[args]]/;! (OrderedQ[args]),
	TVI[depp__, args_List]:>TVI[depp, 
                  Sort[args]]/;! (OrderedQ[args]),
	TJI[depp__, args_List]:>TJI[depp, 
                  Sort[args]]/;! (OrderedQ[args]),
			TBI[depp__, args_List]:>TBI[depp, 
                  Sort[args]]/;! (OrderedQ[args])}

Interchange[z_, i_ ↔ j_, vw__] := 
      Interchange[Interchange[z, i ↔ j], vw];

Interchange[z_, {i_ ↔ j_}] := 
      Interchange[Interchange[z, i ↔ j]];

Interchange[z_, {i_ ↔ j_, vw__}] := 
      Interchange[Interchange[z, i ↔ j], vw];

Interchange[
      test[(Subscript[Δ, x, b, y]), (Subscript[ν, x]), (Subscript[ν, y]), (Subscript[ν, a]), (Subscript[ν, b]), (Subscript[m, x])],  x ↔ y, a ↔ b]

νexplicit

\(νexplicit[
        z_] := \(z /. Σ\_i_ :> \((Expand[
                3\ d\  - 2\  - 
                  2\ \(∑\_\(j = 1\)\%i ν\_j\)])\)\) /. 
	{	σ\_\(i_, j_, k_\) :> \(-\(1\/4\)\) \((d - ν\_i - 
                    2  ν\_j)\) ϕ\_\(i, j, k\) - \(1\/4\) \((d - 
                    2  ν\_i - ν\_j)\) ϕ\_\(j, i, k\) - 
					\(1\/4\) \((2  d - 2  ν\_i - 2  ν\_j - ν\_k - 
                    1)\) ϕ\_\(k, i, j\), 
			h\_\(\(i_\)\(,\)\(j_\)\(,\)\(k_\)\(\ \)\) :> 
				\(-\(1\/2\)\) \((d - 
                    2  ν\_j - ν\_k)\) \(m\_k\^2\) ϕ\_\(i, j, k\) - \(1\/2\) \((2  d - ν\_i - 2  ν\_j - 2  ν\_k - 
                    1)\) \(m\_i\^2\) ϕ\_\(k, i, j\) + 
					\((d - ν\_j - 2\ ν\_k)\) ρ\_\(i, j, k\), 

          S\_\(\(i_\)\(,\)\(j_\)\(,\)\(k_\)\(\ \)\) :> 
				\(-\((d - 
                      2  ν\_j - ν\_k)\)\) \(m\_k\^2\) ϕ\_\(j, i, k\)\  - \((d - ν\_j - 
                    2  ν\_k)\) \(m\_j\^2\) ϕ\_\(k, i, j\) + 
					2 \((2  d - ν\_i - 2  ν\_j - 2  ν\_k - 
                    1)\) ρ\_\(i, j, k\)}\)

TA

MakeBoxes[TA[a_, d___Symbol, ijk___Integer][__], fmt_] := 
      SubscriptBox[a, RowBox[{ijk}]]

MakeBoxes[TA[a_, d___Symbol, ijk___Integer, ___Symbol][__], fmt_] := 
      SubscriptBox[a, RowBox[{ijk}]]

TA[args__][
        mar_List] := (TA[args][mar] = ta[args][mar]);

Assume that the last argument of {em__} is always p =(Sqrt[ (p^2)]).

(72)

ta[D, i_Integer, j_Integer, k_Integer][{em__}]:=With[{q = Last[{em}], 
              mi = {em}[[i]], 
              mj = {em}[[j]], 
              mk = {em}[[k]]},Factor[(q^8)-4 (q^6) ((mi^2) + (mj^2) + (mk^2))+(q^4) (6 ((mi^4) + (mj^4) + (mk^4)) + 
                      4 ((mi^2) (mj^2) + (mk^2) (mi^2) + 
                            (mk^2) (mj^2)))-4 (q^2) ((mi^6) + (mj^6) + (mk^6) - 
                      (mi^2) ((mj^4) + (mk^4)) - 
                      (mj^2) ((mi^4) + (mk^4)) - 
                      (mk^2) ((mi^4) + (mj^4)) + 
                      10 (mi^2) (mj^2) (mk^2))+((Cayley[i, j, k][{em}])^2)]];

(73)

ta[ρ, i_Integer, j_Integer, k_Integer][{em__}] := 
      With[{q = Last[{em}], 
          mi = {em}[[i]], 
          mj = {em}[[j]], 
          mk = {em}[[k]]}, 
        Factor[-(q^6) + 3 (q^4) ((mi^2) + (mj^2) + (mk^2)) - 
            (q^2) (3 ((mi^4) + (mj^4) + (mk^4)) + 
                  2 ((mi^2) (mj^2) + (mk^2) (mi^2) + 
                        (mk^2) (mj^2))) +  ((mi^6) + (mj^6) + (mk^6) - 
                (mi^2) ((mj^4) + (mk^4)) - (mj^2) ((mi^4) + (mk^4)) - 
                (mk^2) ((mi^4) + (mj^4)) + 
                10 (mi^2) (mj^2) (mk^2))]];

(74)

ta[ϕ, i_Integer, j_Integer, k_Integer][{em__}] := 
      With[{q = Last[{em}], 
          mi = {em}[[i]], 
          mj = {em}[[j]], 
          mk = {em}[[k]]}, 
        4  Factor[
            (q^4)  + 
              2  (q^2) ((mi^2) - (mj^2) - 
                    (mk^2)) + ((((mj^2) - (mk^2)))^2) + (mi^2) (2 
                       (mj^2) + 2  (mk^2) - 3  (mi^2))]];

(75)

ta[σ, d_, i_Integer, j_Integer, 
          k_Integer, νi_Integer, νj_Integer, νk_Integer][{em__}] := 
      Factor[-(1/4) (d - νi - 2  νj) TA[ϕ, i, j, 
                  k][{em}] -(1/4) (d - 
                    2  νi - νj) TA[ϕ, j, i, 
                    k][{em}] - (1/4) (2  d - 2  νi - 
                2  νj - νk - 1) TA[ϕ, k, i, 
                j][{em}]   ];

(76)

\(\(ta[h, d_, i_Integer, j_Integer, 
          k_Integer, νi_Integer, νj_Integer, νk_Integer]\)[{em__}] := 
	
      With[{q = Last[{em}], 
          mi = {em}[[i]], 
          mj = {em}[[j]], 
          mk = {em}[[k]]}, 
        Factor[\(-\(1\/2\)\) \((d - 
                  2  νj - νk)\) \(mk\^2\) \(TA[ϕ, i, j, 
                  k]\)[{em}] - \(1\/2\) \((2  d - νi - 2  νj - 
                  2  νk - 1)\) \(mi\^2\) \(TA[ϕ, k, i, 
                  j]\)[{em}] + \(\((d - νj - 2\ νk)\) \(\(TA[ρ, 
                  i, j, k]\)[{em}]\)\(\ \)\)\ \ ]]; \)

(79)

ta[S, d_, i_Integer, j_Integer, 
            k_Integer, νi_Integer, νj_Integer, νk_Integer][{em__}]:=
		With[{q = Last[{em}], 
              mi = {em}[[i]], 
              mj = {em}[[j]], 
              mk = {em}[[k]]},Factor[-(d - 
                        2  νj - νk) (mk^2) TA[ϕ, j, i, 
                      k][{em}] -(d - νj - 
                      2  νk)(mj^2)TA[ϕ, k, i, 
                      j][{em}]test
test2(2  d - νi - 2  νj - 2  νk - 
                      1)TA[ρ, i, j, k][{em}]   ]];

CheckRecursion

The following functions check the recurrence relations for J and V (using the final form as generated by MakeFun) via their Mellin-Integral Kernels.

This test is applicable in the general case  and if some of the 
masses vanish, but not for other special cases !!!

CheckTVIRecursion

\(TVIKernel[d_, PP, \ 
        lis_] := 
\(\((\(-PP\))\)\^\(d - n1 - n2 - n3 - 
              n4\)\) \(\((m1\^2\/\(-PP\))\)\^s1\) \(\((m2\^2\/\(-PP\))\)\^s2\) \(\((m3\^2\/\(-PP\))\)\^s3\) \((m4\^2\/\(-PP\))\)\^s4*
        TVIpsi\ [
          d, \ \((lis /. {m1 -> s1, m2 -> s2, m3 -> s3, 
                m4 -> s4})\)]; \)

\(TVIPsi[
        d_, 
	{{n1_, \ s1_}, \ {n2_, s2_}, {n3_, s3_}, {n4_, 
            s4_}}] := \(\((\(-1\))\)\^\(n1 + n2 + n3 + n4 + 
              1\)\) \(\(gamma[\(-s1\)] gamma[d/2 - n1 - s1]\)\/gamma[
              n1]\) \(\(gamma[\(-s2\)] gamma[d/2 - n2 - s2]\)\/gamma[
              n2]\) \(\(gamma[\(-s3\)] gamma[d/2 - n3 - s3]\)\/gamma[
              n3]\) \(\(gamma[\(-s4\)] gamma[n4 + s4]\)\/gamma[n4]\) 
        gamma[n1 + n2 + n3 + n4 + s1 + s2 + s3 + s4 - d]/
          gamma[3  d/2 - n1 - n2 - n3 - n4 - s1 - s2 - s3 - s4] \(gamma[
              n1 + n3 + s1 + s3 - d/2]\/gamma[
              n1 + n3 + n4 + s1 + s3 + s4 - d/2]\) 
        gamma[d - n1 - n3 - n4 - s1 - s3 - s4]\/gamma[d - n1 - n3 - s1 - s3]; \)

\(CheckTVIRecursion[expr_] := \ 
      If[\ $CheckRecursion === True, 
	
        Block[{rel, kern, \ nkern}, 

          rel = \((\((Extract[
                    expr /. TVIC -> TVIkernel, {1, 2, 1, 1, 2}])\) - 
                Extract[expr /. TVI -> TVIkernel, {1, 2, 1, 2}])\); 

          If[\ {TVIkernel} === 
              Union[Head /@ 
                  Cases[rel, ff_[__] /; Context[ff] =!= "System`", 
                    4]], 	

            rel = Collect[Numerator[Together[rel]], \ TVI[__], Factor]; 

            kern = rel /. TVIkernel -> TVIKernel; 

            kern = kern /. {PP -> \(-1\), m1 -> Sqrt[z1], m2 -> Sqrt[z2], 
                  m3 -> Sqrt[z3], m4 -> Sqrt[z4]}; 

            kern = PowerExpand[kern]; 
kern = Expand[kern]; 

            kern = kern /. {\(z1\^\(s1 + w1_. \)\) \(z2\^\(s2 + 
                            w2_. \)\) \(z3\^\(s3 + w3_. \)\) \(z4\^\(s4 + 
                            w4_. \)\) 
                      fac_ :> \((fac /. {s1 -> s1 - w1, s2 -> s2 - w2, 
                          s3 -> s3 - w3, s4 -> s4 - w4})\)}\ ; 

            kern = \(\(kern /. 
                    TVIpsi -> TVIPsi\) /. {gamma[zz_] :> 
                      gamma[Expand[zz]]}\) /. {gamma[zz_ + nn_Integer] :> 
                    gamma[zz] Pochhammer[zz, nn]}; 
				KERN = kern; 

            kern = kern /. {gamma[zz_] :> 1}; 

            kern = \(kern /. {\((\(-1\))\)\^\(nn_ + ww_Integer\) :> \((\(-1\))\)\^ww}\) /. {\((\(-1\))\)\^nn_ :> 1}; 

            nkern = Table[
			kern /. 
                  Thread[{d, \ n1, \ n2, \ n3, n4, s1, s2, \ s3, s4} -> 
                      Table[Random[], {9}]], {3}]; 

            nkern\ \ , 
		Print["Matching Error"]\ ]\ ], 
		If[$Notebooks, Print["Off"]]]\)

CheckTJIRecursion

\(TJIKernel[d_, PP, \ 
        lis_] := 
\(\((\(-PP\))\)\^\(d - n1 - n2 - 
              n3\)\) \(\((m1\^2\/\(-PP\))\)\^s1\) \(\((m2\^2\/\(-PP\))\)\^s2\) \((m3\^2\/\(-PP\))\)\^s3*
        TJIpsi\ [
          d, \ \((lis /. {m1 -> s1, m2 -> s2, m3 -> s3})\)]; \)

\(TJIPsi\ [d_, {{n1_, \ s1_}, \ {n2_, s2_}, {n3_, s3_}}] := 
      Evaluate[TVIPsi[d, 
	{{n1, \ s1}, \ {n2, s2}, {n3, s3}, {0, 0}}]/
          gamma[0]]\)

Clear[CheckTJIRecursion];

\(CheckTJIRecursion[expr_] := \ 
      If[\ $CheckRecursion === True, 
	
        Block[{rel, kern, nkern}, 

          rel = \((\((Extract[
                    expr /. TJIC -> TJIkernel, {1, 2, 1, 1, 2}])\) - 
                Extract[
                  expr /. TJI -> TJIkernel, {1, 2, 1, 2}])\); 
			REL = 
            rel; 			

          If[{TJIkernel} === 
              Union[Head /@ 
                  Cases[rel, ff_[__] /; Context[ff] =!= "System`", 
                    4]]\ , 			

            rel = Collect[Numerator[Together[rel]], \ TJI[__], Factor]; 

            kern = rel /. TJIkernel -> TJIKernel; 

            kern = kern /. {PP -> \(-1\), m1 -> Sqrt[z1], m2 -> Sqrt[z2], 
                  m3 -> Sqrt[z3]}; 
kern = PowerExpand[kern]; 

            kern = Expand[kern]; 

            kern = kern /. {\(z1\^\(s1 + w1_. \)\) \(z2\^\(s2 + 
                            w2_. \)\) \(z3\^\(s3 + w3_. \)\) 
                      fac_ :> \((fac /. {s1 -> s1 - w1, s2 -> s2 - w2, 
                          s3 -> s3 - w3})\)}; \ 

            kern = \(\(kern /. 
                    TJIpsi -> TJIPsi\) /. {gamma[zz_] :> 
                      gamma[Expand[zz]]}\) /. {gamma[zz_ + nn_Integer] :> 
                    gamma[zz] Pochhammer[zz, nn]}; 
\ \ \ \ \ \ KERN = 
              kern; 
kern = kern /. {gamma[zz_] :> 1}; 

            kern = \(kern /. {\((\(-1\))\)\^\(nn_ + ww_Integer\) :> \((\(-1\))\)\^ww}\) /. {\((\(-1\))\)\^nn_ :> 1}; 

            nkern = Table[
			kern /. 
                  Thread[{d, \ n1, \ n2, \ n3, s1, s2, \ s3} -> 
                      Table[Random[], {7}]], {3}]; 

            nkern\ \ , 
		Print["Matching Error"]\ ]\ ], 
		If[$Notebooks, Print["Off"]]]\)

Recurrence Relations Input and Recursion Function Generation

TFR[d_, pp_, {{n1_, m1_}, {n2_, m2_}, {0, 0}, {0, 0}, {0, 0}}] := 
      TAI[d, pp, {{n1, m1}}] TAI[d, {{n2, m2}}]

Most (Subscript[e, ... ])equations below were entered by the palettes which show up when $PutPalettes is set to True.
Examples of alternative (hand) input are  (50) , (201) -- (205).

Input (35)

general (Subscript[m, i]^2)

(Subscript[e, 35])=
2(Subscript[ν, 1])𝒞 \(1\^+\)(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)))== ((d - 
                      2  (Subscript[ν, 1]) - (Subscript[ν, 3]) - (Subscript[ν, 5])) (Subscript[Δ, 1])+(Subscript[Δ, 3, 4, 5])((Subscript[ν, 5])\(5\^+\)(\(2\^-\)-\(1\^-\))-(Subscript[ν, 3])\(3\^+\)\(1\^-\))+(Subscript[Δ, 2])((Subscript[ν, 1])\(1\^+\)(\(5\^-\)-\(2\^-\))+(Subscript[ν, 3])\(3\^+\)(\(5\^-\)-\(4\^-\))+(Subscript[ν, 5])-(Subscript[ν, 1]))+
				(Subscript[Δ, 6])((Subscript[ν, 1])\(1\^+\)\(3\^-\)+(Subscript[ν, 5])\(5\^+\)(\(3\^-\)-\(4\^-\))+(Subscript[ν, 3])-(Subscript[ν, 1])))(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)));

(Subscript[s, 35]) = OperatorApplyF[(Subscript[e, 35])]

MakeFun (35)

(Subscript[h, 351]) = (Subscript[s, 35]) /. (Subscript[ν, 1]) -> ((Subscript[ν, 1]) - 1)

(1^(+ )) (35_1)

(Subscript[f, 351])=MakeFun[(Subscript[h, 351]),"(35_1)",IFF[(Subscript[ν, 1]) > 1&&Cayley[(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),(Sqrt[(p^2)])]=!=0]]

(2^(+ )) (35_2)

(Subscript[f, 352])=MakeFun[Interchange[(Subscript[h, 351]), 1 ↔ 2, 
            3 ↔ 4],"(35_2)",IFF[(Subscript[ν, 2]) > 1&&Cayley[(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),(Sqrt[(p^2)])]=!=0]]

(3^(+ )) (35_3)

(Subscript[f, 353])=MakeFun[Interchange[(Subscript[h, 351]), 1 ↔ 3, 
            2 ↔ 4],"(35_3)",IFF[(Subscript[ν, 3]) > 1&&Cayley[(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),(Sqrt[(p^2)])]=!=0]]

(4^(+ )) (35_4)

(Subscript[f, 354])=MakeFun[Interchange[(Subscript[h, 351]), 1 ↔ 4, 
            2 ↔ 3],"(35_4)",IFF[(Subscript[ν, 4]) > 1&&Cayley[(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),(Sqrt[(p^2)])]=!=0]]

Input (3501)

Cayley[m1, m5, m5, m4, m5, m5] // Factor

(Subscript[m, 1]) = 0  ;   (Subscript[m, 2]) = (Subscript[m, 3]) = (Subscript[m, 5]);    (p^2) = 
      (Subscript[m, 5]^2)

(Subscript[e, 3501])=(d - 2  (Subscript[ν, 1]) - (Subscript[ν, 3]) - (Subscript[ν, 5]))(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)))==( (Subscript[ν, 5])(\(1\^-\)-\(2\^-\))\(5\^+\)+(Subscript[ν, 3])\(1\^-\)\(3\^+\) )(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)));

(Subscript[s, 3501]) = OperatorApplyF[(Subscript[e, 3501])]

MakeFun (3501) Δ=0 ((p^2) = (m^2))

(Subscript[h, 3501]) = (Subscript[s, 3501])

(Subscript[f, 3501])=MakeFun[(Subscript[s, 3501]),"(3501)",IFF[PP === m5^2],{m1 :> 0, m2 :> m5, m3 :> m5}]

Input (3502)

Cayley[m1, m5, m5, m4, m5, m5] // Factor

(Subscript[m, 4]) = 0  ;   (Subscript[m, 2]) = (Subscript[m, 3]) = (Subscript[m, 5]);    (p^2) = 
      (Subscript[m, 5]^2)

(Subscript[e, 3502])=(d - 2  (Subscript[ν, 1]) - (Subscript[ν, 3]) - (Subscript[ν, 5]))(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)))==( (Subscript[ν, 5])(\(1\^-\)-\(2\^-\))\(5\^+\)+(Subscript[ν, 3])\(1\^-\)\(3\^+\) )(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)));

(Subscript[s, 3502]) = OperatorApplyF[(Subscript[e, 3502])]

MakeFun (3502) Δ=0 ((p^2) = (m^2))

(Subscript[h, 3502]) = (Subscript[s, 3502])

(Subscript[f, 3502])=MakeFun[(Subscript[h, 3502]),"(3502)",IFF[PP === m5^2],{m4 :> 0, m2 :> m5, m3 :> m5}]

MakeFun (3503) Δ=0 ((p^2) = (m^2))

Cayley[m5, 0, m3, m5, m5, m5]

(Subscript[h, 3503]) = 
      Interchange[(Subscript[s, 3502]), 1 ↔ 2,  
        3 ↔ 4]

(Subscript[f, 3503])=MakeFun[(Subscript[h, 3503]),"(3503)",IFF[PP === m5^2],{m2 :> 0, m1 :> m5, m4 :> m5}]

MakeFun (3504) Δ=0 ((p^2) = (m^2))

Cayley[m5, m2, 0, m5, m5, m5]

(Subscript[h, 3504]) = 
      Interchange[(Subscript[s, 3502]), 1 ↔ 3,  
        2 ↔ 4]

(Subscript[f, 3504])=MakeFun[(Subscript[h, 3504]),"(3504)",IFF[PP === m5^2],{m3 :> 0, m1 :> m5, m4 :> m5}]

Input (41)

general (Subscript[m, i]^2)

(Subscript[e, 41])=
2(Subscript[ν, 5]) 𝒞 \(5\^+\)(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d))) == ((Subscript[Δ, 1, 3, 6])((Subscript[ν, 3])\(3\^+\)(\(4\^-\)-\(5\^-\))+(Subscript[ν, 1])\(1\^+\)(\(2\^-\)-\(5\^-\)))+(d - (Subscript[ν, 1]) - (Subscript[ν, 3]) - 
                      2  (Subscript[ν, 5])) (Subscript[Δ, 5])+(Subscript[Δ, 2])((Subscript[ν, 5])\(5\^+\)(\(1\^-\)-\(2\^-\))+(Subscript[ν, 3])\(3\^+\)\(1\^-\)+(Subscript[ν, 1])-(Subscript[ν, 5]))+(Subscript[Δ, 4])((Subscript[ν, 1])\(1\^+\)\(3\^-\)+(Subscript[ν, 5])\(5\^+\)(\(3\^-\)-\(4\^-\))+(Subscript[ν, 3])-(Subscript[ν, 5])))(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)));

(Subscript[s, 41]) = OperatorApplyF[(Subscript[e, 41])]

MakeFun (41)

(Subscript[h, 41]) = (Subscript[s, 41]) /. (Subscript[ν, 5]) -> (Subscript[ν, 5]) - 1

(Subscript[f, 41])=MakeFun[(Subscript[h, 41]),"(41)",IFF[(Subscript[ν, 5]) > 1&&Cayley[(Subscript[m, 1]),(Subscript[m, 2]),(Subscript[m, 3]),(Subscript[m, 4]),(Subscript[m, 5]),(Sqrt[(p^2)])]=!=0]]

Input (43)

general (p^2) and ( (Subscript[m, 5]^2) == 0  &&  (Subscript[m, 1]^2)==(Subscript[m, 2]^2) && (Subscript[m, 3]^2)== (Subscript[m, 4]^2) ) ⟹ Δ == 0

(Subscript[e, 43])=(d - (Subscript[ν, 1]) - (Subscript[ν, 3]) - 2  (Subscript[ν, 5]))(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)))==((Subscript[ν, 1])\(1\^+\)(\(5\^-\)-\(2\^-\))+(Subscript[ν, 3])\(3\^+\)(\(5\^-\)-\(4\^-\)))(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)));

(Subscript[s, 43]) = OperatorApplyF[(Subscript[e, 43])]

Input (44)

general (p^2) and ( (Subscript[m, 5]^2) == 0  &&  (Subscript[m, 1]^2)==(Subscript[m, 2]^2) && (Subscript[m, 3]^2)== (Subscript[m, 4]^2) ) ⟹ Δ == 0

(Subscript[e, 44])=(d - (Subscript[ν, 2]) - (Subscript[ν, 4]) - 2  (Subscript[ν, 5]))(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)))==((Subscript[ν, 2])\(2\^+\)(\(5\^-\)-\(1\^-\))+(Subscript[ν, 4])\(4\^+\)(\(5\^-\)-\(3\^-\)))(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)));

(Subscript[s, 44]) = OperatorApplyF[(Subscript[e, 44])]

Input (45)

(Subscript[e, 45])= (Subscript[Δ, 1, 3, 6])\(1\^+\)(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)))==((Subscript[u, 6, 1, 3])((Subscript[ν, 1])\(1\^+\)\(3\^-\)+(Subscript[ν, 5])\(5\^+\)(\(3\^-\)-\(4\^-\)))-2(Subscript[u, 1, 3, 6])( (Subscript[ν, 2])\(2\^+\)(\(5\^-\)-\(1\^-\))+ (Subscript[ν, 4])\(4\^+\)(\(5\^-\)-\(3\^-\))+-(Subscript[Σ, 5])-(Subscript[ν, 1])+3  (Subscript[ν, 3]))+2(Subscript[m, 3]^2)((Subscript[ν, 5])\(5\^+\)(\(1\^-\)-\(2\^-\))+(Subscript[ν, 3])\(3\^+\)\(1\^-\)+(Subscript[ν, 1])-(Subscript[ν, 3])))(Subscript[F, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])(Subscript[ν, 5])]^((d)));

(Subscript[s, 45]) = OperatorApplyF[(Subscript[e, 45])]

MakeFun (43) and (44)

(Subscript[f, 43]) = 
      MakeFun[(Subscript[s, 43]), "(43)", 
        IFF[(Subscript[ν, 1]) + (Subscript[ν, 3]) + 2  (Subscript[ν, 5]) - (d /. dd_Symbol :> 4) =!= 
            0], {m5 :> 0, m4 :> m3, m2 :> m1}]

(Subscript[f, 44]) = 
      MakeFun[(Subscript[s, 44]), "(44)", {m5 :> 0, m4 :> m3, m2 :> m1}]

MakeFun (50)

Subscript[f, 50] = 
   HoldForm[TFR[d_, PP_, 
      {{(n1_)?PQ, m1_}, {(n2_)?PQ, m2_}, {(n3_)?PQ, m3_}, 
       {(n4_)?PQ, m4_}, {(n5_)?PQ, m5_}}] := 
     (TComment["(50)", TFIC[d, PP, 
          {{n1, m1}, {n2, m2}, {n3, m3}, {n4, m4}, 
           {n5, m5}}]]; 
        Block[{r1 = 3 - d, r2 = 4 - d, 
          r3 = 
           -(m1^2*m2^2*m3^2) + m2^4*m3^2 + m2^2*m3^4 + 
            m1^4*m4^2 - m1^2*m2^2*m4^2 - 
            m1^2*m3^2*m4^2 - m2^2*m3^2*m4^2 + 
            m1^2*m4^4 + m1^2*m2^2*m5^2 - 
            m2^2*m3^2*m5^2 - m1^2*m4^2*m5^2 + 
            m3^2*m4^2*m5^2 + m1^2*m3^2*PP - 
            m2^2*m3^2*PP - m1^2*m4^2*PP + m2^2*m4^2*PP - 
            m1^2*m5^2*PP - m2^2*m5^2*PP - m3^2*m5^2*PP - 
            m4^2*m5^2*PP + m5^4*PP + m5^2*PP^2, 
          r4 = -4 + d, r5 = -3 + d, r6 = -7 + 2*d, 
          r7 = -(m2^2*r4) + m3^2*r5 - m5^2*r5 + m4^2*r6, 
          r8 = -10 + 3*d, r9 = -m3^2 + m4^2, 
          r10 = -11 + 3*d, 
          r11 = -(m4^2*r10) - 3*m5^2*r5 + m3^2*r6, 
          r12 = 
           m2^2*r11 + m5^4*r5 + m2^4*r8 + m4^2*r4*r9 + 
            m5^2*r5*r9, r13 = -6 + d, 
          r14 = -(m4^2*r13) + m5^2*r5, 
          r15 = -3*m3^2 - m4^2*r10 + m5^2*r5, 
          r16 = 
           m3^2*r14 + m2^2*r15 + m2^4*r4 - 2*m3^4*r5 - 
            3*m4^2*m5^2*r5 + m5^4*r5 + m4^4*r8, 
          r17 = m3^2 + 2*m4^2, r18 = -1 + d, 
          r19 = -(m3^2*r13) - 2*m4^2*r18 + 2*m5^2*r5, 
          r20 = 
           2*m2^4 - 3*m3^4 + d*m3^4 - 3*m3^2*m4^2 + 
            2*m4^4 + m2^2*r19 - 2*m5^4*r5 + m5^2*r17*r5, 
          r21 = 2 + d, r22 = 
           2*m5^2 - m4^2*r21 + m2^2*r4 - m3^2*r4, 
          r23 = -7 + d, r24 = 6*m4^2 - 4*m5^2, 
          r25 = 3*m3^2 + m4^2*r18 + m5^2*r23, 
          r26 = 
           -6*m4^4 - 2*m4^2*m5^2*r23 + m3^2*r24 + 
            2*m2^2*r25 - 2*m2^4*r4 - m3^4*r4, 
          r27 = m3^4 + m3^2*m4^2 - 2*m4^4, 
          r28 = m3^2 + m4^2*r23, 
          r29 = -2*m4^2*r18 + m3^2*r21 + 2*m5^2*r23, 
          r30 = -6*m2^4 + 2*m5^2*r28 - m2^2*r29 + r27*r4, 
          r31 = m1^4*r22 + m1^2*r26 + m3^2*r30 + m1^6*r4, 
          r32 = m3 - m4, r33 = m3 + m4, 
          r34 = 2*m3^2 + m4^2, r35 = -17 + 3*d, 
          r36 = m3^2*r23 + m4^2*r35, r37 = -9 + d, 
          r38 = 17 - 3*d, 
          r39 = 5*m3^2 + m4^2*r37 + m5^2*r38, 
          r40 = 1 + d, r41 = 
           5*m4^2 - m5^2*r23 + m2^2*r4 - m3^2*r40, 
          r42 = 
           m5^4*r35 - m5^2*r36 + m2^2*r39 - 2*m1^4*r4 + 
            m2^4*r4 - r32*r33*r34*r4 + m1^2*r41, 
          r43 = m1^2 - m2^2 + m3^2 - m4^2, 
          r44 = 2*m5^2*r13 + r4*r43, 
          r45 = 
           m3^4*r12 + m1^4*r16 + m1^2*m3^2*r20 + 
            PP*r31 + PP^2*r42 + PP^3*r44 + m1^6*r7, 
          r46 = m2^2 + m4^2, r47 = 13 - 3*d, 
          r48 = m4^2*r47 - m5^2*r5 + m3^2*r6, 
          r49 = m5^4*r5 + m5^2*r32*r33*r5 + m4^2*r4*r9, 
          r50 = -18 + 4*d, r51 = -(m4^2*r13) - 3*m5^2*r5, 
          r52 = 
           2*m3^4*r5 + m4^2*m5^2*r5 + m5^4*r5 + 
            m4^4*r50 + m3^2*r51, 
          r53 = -3*m4^2 + 2*m5^2*r5, 
          r54 = 
           m4^4*r47 + 2*m3^4*r5 + m4^2*m5^2*r5 - 
            2*m5^4*r5 + m3^2*r53, 
          r55 = m4^2*r13 + 6*m3^2*r5 - 2*m5^2*r5, 
          r56 = 3*m4^2 + m3^2*r5 - m5^2*r5, 
          r57 = 7 - 2*d, 
          r58 = m3^2*r5 + 3*m5^2*r5 + m4^2*r57, 
          r59 = 
           m2^6*r4 + m2^2*m4^2*r55 + m2^4*r56 + m4^4*r58, 
          r60 = m4^2 - m3^2*r5, r61 = 10 - 3*d, 
          r62 = 2*m5^2 + 3*m4^2*r4 + m3^2*r61, 
          r63 = m4^2 + m5^2*r5, 
          r64 = 
           -4*m4^2*m5^2 + 3*m4^4*r4 - 2*m3^4*r5 + 
            2*m3^2*r63, r65 = m4^2 + m3^2*r5 - m5^2*r5, 
          r66 = 2*m3^2*r5 + 2*m5^2*r5 + m4^2*r61, 
          r67 = 3*m2^4*r4 + 2*m2^2*r65 + m4^2*r66, 
          r68 = 
           -3*m2^6*r4 - 2*m1^4*m4^2*r5 + 
            2*m4^2*m5^2*r60 + m2^4*r62 + m2^2*r64 + 
            m1^2*r67 - 3*m4^4*r4*r9, r69 = -13 + 3*d, 
          r70 = m3^2*r5 + m4^2*r69, 
          r71 = m4^2 - 3*m2^2*r4 - m3^2*r5 + m5^2*r5, 
          r72 = -5 + d, r73 = m3^2 + m5^2*r69 + m4^2*r72, 
          r74 = 
           3*m2^4*r4 - m5^4*r5 + m5^2*r70 + m1^2*r71 + 
            m2^2*r73 + 3*m4^2*r4*r9, 
          r75 = m1^2 - m2^2 + m3^2 - m4^2 - 2*m5^2, 
          r76 = 
           m2^8*r4 + m2^6*r48 + m4^4*r49 + 
            2*m1^4*m4^2*r46*r5 + m2^4*r52 + 
            m2^2*m4^2*r54 - m1^2*r59 + PP*r68 + 
            PP^2*r74 + PP^3*r4*r75, 
          r77 = m3^2 - m4^2 + m5^2, 
          r78 = -2*m2^2 + m4^2 + m5^2, 
          r79 = -2*m1^2 - m3^2 - m4^2 + m5^2, 
          r80 = 
           m1^4 + PP^2 - m1^2*r77 + m3^2*r78 + PP*r79, 
          r81 = m3^2 + m4^2, r82 = m3^2 + m4^2 + m5^2, 
          r83 = m1^2 + m3^2 - m5^2, 
          r84 = 
           m1^4 + 2*m3^2*m4^2 - m4^4 + PP^2 - m2^2*r77 + 
            m5^2*r81 - m1^2*r82 - 2*PP*r83, 
          r85 = m1 - m3, r86 = -PP + r85^2, 
          r87 = m1 + m3, r88 = -PP + r87^2, 
          r89 = m2 - m4, r90 = m2 + m4, 
          r91 = -m1^2 - m2^2 - m3^2 - m4^2 + 2*m5^2, 
          r92 = PP^2 + r85*r87*r89*r90 + PP*r91, 
          r93 = m2^2 - 3*m4^2, r94 = m4 - m5, 
          r95 = m4 + m5, r96 = 2*m3^2 - m4^2 + m5^2, 
          r97 = -m1^2 - m2^2 - 2*m3^2 + 3*m5^2, 
          r98 = 
           PP^2 + m1^2*r93 + m2^2*r94*r95 + m4^2*r96 + 
            PP*r97, r99 = -2 + d, r100 = -m1^2 + m2^2, 
          r101 = m2^2 + m1^2*r72, 
          r102 = m3^2*r101 - 2*m3^4*r4 + m2^2*r100*r4, 
          r103 = 9 - 2*d, r104 = m2^2*r23 + m3^2*r72, 
          r105 = 
           2*m1^4 + m2^4*r103 + m1^2*r104 + m2^2*m3^2*r5, 
          r106 = -9 + 2*d, r107 = m1 - m2, 
          r108 = m1 + m2, 
          r109 = -(r106*r107*r108) + m3^2*r4, 
          r110 = 
           m2^2*r102 + m4^2*r105 + m4^4*r109 - m4^6*r4, 
          r111 = m1^2 + m2^2, 
          r112 = m2^2*r4 - 2*m1^2*r72, r113 = -19 + 5*d, 
          r114 = m3^2*r112 + m3^4*r113 + m2^2*r111*r4, 
          r115 = m3^2*r5 + m2^2*r6, 
          r116 = 
           2*m1^4 + 2*m1^2*r115 + m2^4*r13 + m3^4*r72 - 
            2*m2^2*m3^2*r99, 
          r117 = m2^2*r106 + m3^2*r4 - m1^2*r99, 
          r118 = 
           m2^2*r114 - m4^2*r116 + m4^4*r117 - m4^6*r6, 
          r119 = -(m2^2*r5) + 2*m3^2*r57 + m1^2*r72, 
          r120 = m1^2*r18 + 2*m3^2*r4 - m2^2*r5, 
          r121 = m2^2*r119 + m4^2*r120 + 2*m4^4*r5, 
          r122 = m3^2*r5 + m4^2*r72, 
          r123 = -2*m2^2*r4 - m4^2*r69 + m3^2*r72, 
          r124 = 
           m2^2*r122 + m1^2*r123 + 2*m2^4*r4 + 
            2*m4^2*r32*r33*r4, 
          r125 = -3*m4^2 + m2^2*r4 - m3^2*r72, 
          r126 = m3^2 - m4^2*r72, 
          r127 = 
           -2*m3^2*m4^2 - 2*m1^2*r125 - 2*m2^2*r126 + 
            m4^4*r18 - 4*m2^4*r4 + m3^4*r72, 
          r128 = 
           -(m2^2*r18) + 2*m4^2*r5 + m1^2*r72 + 
            2*m3^2*r72, 
          r129 = 
           m5^2*r127 - m5^4*r128 - r124*r32*r33 + 
            m5^6*r72, r130 = -m1^2 + m2^2 + m3^2 - m4^2, 
          r131 = -14 + 3*d, 
          r132 = 
           -(m3^2*r13) + m4^2*r131 + m1^2*r4 + 5*m2^2*r4, 
          r133 = -2*m5^4 + m5^2*r132 + r130*r32*r33*r4, 
          r134 = 
           m5^2*r118 + m5^4*r121 + PP*r129 + PP^2*r133 + 
            r110*r32*r33 - 2*m5^2*PP^3*r4 + 
            m5^6*r5*r89*r90, r135 = m3 - m4 - m5, 
          r136 = m3 + m4 - m5, r137 = m3 - m4 + m5, 
          r138 = m3 + m4 + m5, r139 = -10 + d, 
          r140 = 13 - 2*d, r141 = d*m4^2 + 2*m3^2*r5, 
          r142 = -8 + d, 
          r143 = m4^2*r142 - m3^2*r99 + m5^2*r99, 
          r144 = 
           m3^2*m4^2*r139 + m4^4*r140 - m5^2*r141 + 
            m2^2*r143 + m3^4*r5 + m5^4*r5, 
          r145 = 3*m3^4 - 12*m3^2*m4^2 + 7*m4^4, 
          r146 = 3*m3^2 + 4*m4^2, r147 = m3^2 - 11*m4^2, 
          r148 = 
           7*m3^4 + 10*m3^2*m4^2 - 17*m4^4 - 5*m5^4 - 
            2*m5^2*r147, r149 = m2^2 + 2*m3^2 + m4^2, 
          r150 = 
           -m2^4 + m2^2*m3^2 + m3^4 + 3*m2^2*m4^2 + 
            m3^2*m4^2 - m4^4 + m5^4 - m5^2*r149, 
          r151 = 
           3*m3^6 + 6*m3^4*m4^2 - 17*m3^2*m4^4 + 
            8*m4^6 + 3*m5^6 - m5^2*r145 - m5^4*r146 + 
            m2^2*r148 - d*r150*r77 + 6*m2^4*r94*r95, 
          r152 = -(m3^2*m4) + m4^3, r153 = 11 - 2*d, 
          r154 = -(d*m3^2*m4^2) + m4^4*r153 + m3^4*r5, 
          r155 = m4^2 + m3^2*r5, 
          r156 = r142*r32*r33 + m5^2*r99, 
          r157 = 11 - 3*d, r158 = -8 + 3*d, 
          r159 = m5^2*r158 + m4^2*r21, 
          r160 = 
           -3*m3^4 + 2*m4^2*m5^2*r106 + m5^4*r157 + 
            m3^2*r159 - m4^4*r18, 
          r161 = 
           -2*m2^6 + m5^2*r154 - 2*m5^4*r155 + 
            m2^4*r156 + m2^2*r160 + r152^2*r4 + m5^6*r5, 
          r162 = m4^2*r23 - 3*m3^2*r72, 
          r163 = -4*m2^2 + m4^2*r18 + m3^2*r23, 
          r164 = -m3^4 + m4^4, 
          r165 = 
           -(m2^4*r13) + m2^2*r162 + m1^2*r163 + 
            r164*r4 + m1^4*r99, r166 = -19 + 4*d, 
          r167 = m4^2 + m3^2*r72, r168 = m3^2 + m4^2*r69, 
          r169 = -2*r168 + 4*m2^2*r4, 
          r170 = 
           -m3^4 + m2^4*r13 - m4^4*r166 + 2*m2^2*r167 + 
            m1^2*r169 + 2*m3^2*m4^2*r72 + m1^4*r99, 
          r171 = m1^2 - m2^2 + 4*m4^2, 
          r172 = 
           -(m5^2*r170) + r165*r32*r33 + m5^6*r72 - 
            m5^4*r171*r72, r173 = m2^2 + m3^2, 
          r174 = -(m4^2*r13) + r131*r173 + m1^2*r8, 
          r175 = m5^2*r174 - 2*m5^4*r4 + r130*r32*r33*r4, 
          r176 = 
           2*m1^6*m4^2 + m1^4*r144 + m1^2*r151 + 
            m3^2*r161 + PP*r172 + PP^2*r175 - 
            2*m5^2*PP^3*r4, r177 = m3^2 + m4^2 - m5^2, 
          r178 = -2*m1^2 + m3^2 - m4^2 + m5^2, 
          r179 = -m3^2 + m4^2 + m5^2, 
          r180 = m2^2*r177 + m4^2*r178 + PP*r179, 
          r181 = -2*m2^2 - m3^2 + m4^2 + m5^2, 
          r182 = m1^2*r177 + m3^2*r181 + PP*r77, 
          r183 = -(m2^2*r13) - 2*m3^2*r5, 
          r184 = 
           m1^2*r183 - 2*m3^4*r4 + m2^4*r57 + m1^4*r72 - 
            m2^2*m3^2*r99, r185 = 3*m2^2*r4 + m3^2*r72, 
          r186 = 
           m2^4*r106 + 2*m1^2*r185 + m1^4*r40 - 
            2*m2^2*m3^2*r6, r187 = r111*r13 - m3^2*r4, 
          r188 = 
           m2^2*r184 + m4^2*r186 - m4^4*r187 + m4^6*r4, 
          r189 = 2*m1^2 + m3^2*r10 + 2*m2^2*r5, 
          r190 = m2^2*r5 + m3^2*r72 + 2*m1^2*r99, 
          r191 = m2^2*r189 - m4^2*r190 - m4^4*r5, 
          r192 = -(m3^2*r5) - m4^2*r72, 
          r193 = 2*m2^2*r4 + m4^2*r69 - m3^2*r72, 
          r194 = 
           m2^2*r192 + m1^2*r193 - 2*m2^4*r4 + 
            2*m4^2*r4*r9, r195 = m2^2 + m4^2 + m3^2*r72, 
          r196 = -(m3^2*r6) - m4^2*r72, 
          r197 = 
           m2^4*r157 + 2*m1^2*r195 + 2*m2^2*r196 + 
            2*m4^2*r17*r4 + m1^4*r72, 
          r198 = 
           m4^2*r10 - 2*m2^2*r5 + 2*m1^2*r72 + m3^2*r72, 
          r199 = 
           r107*r108*r194 + m5^2*r197 - m5^4*r198 + 
            m5^6*r72, r200 = m1^2 - m2^2 - m3^2 + m4^2, 
          r201 = m1^2 + m2^2 + m3^2 + 5*m4^2, 
          r202 = 2*m5^4 - r107*r108*r200 - m5^2*r201, 
          r203 = 
           r107*r108*r110 - m5^2*r188 - m5^4*r191 + 
            PP*r199 + 2*m5^2*PP^3*r4 + PP^2*r202*r4 + 
            m5^6*r5*r89*r90, r204 = m4^2 + m5^2, 
          r205 = -2*m5^2*r5 + m4^2*r99, r206 = 18 - 5*d, 
          r207 = m4^2*r206 + m5^2*r8 - 2*m3^2*r99, 
          r208 = 
           m2^2*r207 + 2*m2^4*r4 - m3^4*r5 - 
            m3^2*r204*r5 + r205*r94*r95, 
          r209 = 2*m3^2*m4^2 + m4^4*r142 + 2*m3^4*r5, 
          r210 = -(m4^2*r18) + m3^2*r5, r211 = 3 - 2*d, 
          r212 = m5^2 + m4^2*r157 + m3^2*r211, 
          r213 = m4^2*r106 + m3^2*r4, 
          r214 = 
           -(d*m3^4) + 10*m3^2*m4^2 - 2*d*m3^2*m4^2 - 
            2*m4^4 + d*m4^4 + 2*m5^2*r213 - 2*m5^4*r4, 
          r215 = 
           d*m3^2*m4^4 - 2*m4^6 - m5^2*r209 + 
            m5^4*r210 + m2^4*r212 + m2^2*r214 + 
            m2^6*r4 + m5^6*r5 - m3^4*m4^2*r99, 
          r216 = m4^2*r13 + m3^2*r99, 
          r217 = m3^2*r5 - m4^2*r72, r218 = 13 - 4*d, 
          r219 = 3*m3^2 + m5^2*r218 + m4^2*r23, 
          r220 = m4^2*r142 + m5^2*r61, 
          r221 = 
           2*m3^4 + 2*m4^2*m5^2 - m4^4*r13 + m3^2*r220 + 
            2*m5^4*r6, r222 = 
           m4^2*m5^2*r216 + m5^4*r217 + m2^4*r219 + 
            m2^2*r221 - m5^6*r5 + m2^6*r99, 
          r223 = -(m3^2*r10) + m4^2*r5, 
          r224 = 
           4*m3^2*m4^2 + m1^2*r122 + m4^4*r13 + 
            m2^2*r223 + m1^4*r4 - m2^4*r4 - m3^4*r99, 
          r225 = -m3^2 + m2^2*r72 - m4^2*r72, 
          r226 = -2*m4^2*r106 + 2*m3^2*r69, 
          r227 = 
           m1^4*r106 + m2^4*r106 - m4^4*r13 - 
            2*m1^2*r225 + m2^2*r226 - 4*m3^2*m4^2*r4 - 
            m3^4*r99, r228 = m3^2 + 3*m4^2, 
          r229 = 
           r107*r108*r224 - m5^2*r227 + m5^6*r72 - 
            m5^4*r228*r72, 
          r230 = m4^2*r131 + m1^2*r4 + m2^2*r4 + m3^2*r8, 
          r231 = 
           2*m5^4*r106 - m5^2*r230 - r107*r108*r200*r4, 
          r232 = 
           m1^4*r208 - m1^2*r215 - m3^2*r222 + PP*r229 + 
            PP^2*r231 + 2*m5^2*PP^3*r4 + m1^6*r7, 
          r233 = -2*m3^2 + m4^2, 
          r234 = -m2^2 + m4^2 - 2*m5^2, 
          r235 = -m1^2 + m2^2 + m5^2, 
          r236 = 
           m1^4 + m5^4 + m2^2*r233 + m1^2*r234 + 
            PP*r235 - m5^2*r46, r237 = -m2^2 + m5^2, 
          r238 = m2^2 + m3^2 - 2*m4^2 + m5^2, 
          r239 = -m1^2 + m2^2 - m5^2, 
          r240 = m1^4 + m3^2*r237 - m1^2*r238 + PP*r239, 
          r241 = m3^2 + 3*m4^2 - m5^2, 
          r242 = -3*m3^2 - m4^2 + m5^2, 
          r243 = 
           m5^4 - m5^2*r228 + m1^2*r241 + m2^2*r242 + 
            2*PP*r32*r33 + 2*m4^2*r9, 
          r244 = 2*m1^2 + m2^2 + m4^2, 
          r245 = -2*m1^2 - m3^2 - m4^2 + 3*m5^2, 
          r246 = 
           m1^4 - 3*m2^2*m3^2 + m5^4 + PP^2 + 
            m4^2*r173 - m5^2*r244 + PP*r245, 
          r247 = -2*m3^2 + m4^2 + m5^2, 
          r248 = 
           -m2^4 - m4^2*m5^2 + PP*r235 + m2^2*r247 + 
            m1^2*r46, r249 = m2^2*r153 + m1^2*r5, 
          r250 = 2*m1^2 + m2^2*r4, 
          r251 = m2^2*r13 + m1^2*r139, 
          r252 = r107*r108*r250 + m3^2*r251 - m3^4*r4, 
          r253 = m1^2*r140 + 2*m3^2*r4 + m2^2*r72, 
          r254 = 
           m3^4*r249 + m4^2*r252 + m4^4*r253 - m4^6*r4 + 
            m2^2*m3^2*r100*r99, r255 = 2*m3^2 + m4^2*r13, 
          r256 = -2*m3^2*r5 + m2^2*r99 - m4^2*r99, 
          r257 = -11 + 2*d, 
          r258 = 
           -(m4^4*r106) + 2*m3^2*m4^2*r23 - m3^4*r257, 
          r259 = 4*m4^2 + m3^2*r13, 
          r260 = 
           -2*m2^4 + 6*m3^2*m4^2 + 2*m2^2*r259 + 
            m4^4*r37 + m3^4*r5, 
          r261 = 
           -5*m3^4*m4^2 + m2^4*r255 + m1^4*r256 + 
            m2^2*r258 + m1^2*r260 - m2^6*r4 + m3^6*r5 + 
            m4^6*r6 - 3*m3^2*m4^4*r72, 
          r262 = m3^4 + m3^2*m4^2 + m4^4, 
          r263 = 
           m1^4 + m2^4 - 2*r262 + m1^2*r81 + m2^2*r81, 
          r264 = m1^2 + m2^2 - m3^2 - m4^2, 
          r265 = -(r13*r32*r33) + m2^2*r99, 
          r266 = 
           -2*m1^2*r265 - r32^2*r33^2*r4 + 
            2*m2^2*r13*r9 + m1^4*r99 + m2^4*r99, 
          r267 = m3^2*r13 + m4^2*r4, r268 = -16 + 3*d, 
          r269 = m3^2*r13 - m4^2*r268 + m2^2*r4, 
          r270 = 
           2*m2^2*r267 + 2*m1^2*r269 + 
            r13*r228*r32*r33 + m2^4*r61 + m1^4*r99, 
          r271 = 2*m2^2 + m4^2*r23 - m3^2*r72, 
          r272 = 
           -(m5^2*r270) - 2*m5^4*r271 + r266*r32*r33, 
          r273 = 2*m3^2 - 2*m4^2 - m5^2, 
          r274 = m1^2*r5 - m2^2*r5 + r273*r72, 
          r275 = 
           r200*r254 + m5^2*r261 + PP*r272 + 
            2*m5^2*PP^2*r274 + m5^4*r263*r5 - 
            m5^6*r264*r5, 
          r276 = 
           -2*m2^2*m3^2 - 2*m3^4 + 3*m3^2*m4^2 - m4^4 + 
            3*m3^2*m5^2 + 2*m4^2*m5^2 - m5^4 + 
            m1^2*r177 + PP*r77, 
          r277 = -(m4^2*r106) + m2^2*r4 - m3^2*r5, 
          r278 = 
           -2*m3^4 - m3^2*m4^2*r13 + m2^2*r196 + 
            m2^4*r4 + m4^4*r4, 
          r279 = m4^2*r131 + m3^2*r8, 
          r280 = 
           m2^2*r279 - 2*m2^4*r4 + m4^2*r32*r33*r99, 
          r281 = m1^4*r277 + m2^2*r278 + m1^2*r280, 
          r282 = 11*m3^2 - 3*d*m3^2 + 9*m4^2 - 2*d*m4^2, 
          r283 = 15 - 4*d, 
          r284 = m4^2*r106 + m2^2*r283 + m3^2*r5, 
          r285 = m4^4*r13 + 4*m3^2*m4^2*r5 - m3^4*r99, 
          r286 = -m4^2 + m3^2*r5, 
          r287 = -(m4^2*r5) + m3^2*r6, 
          r288 = 2*m2^2*r287 + m2^4*r72 - 2*r286*r81, 
          r289 = 
           -2*m3^2*m4^4 + m2^4*r282 + m1^4*r284 + 
            m2^2*r285 + m1^2*r288 - m4^6*r4 + m1^6*r5 + 
            m2^6*r6 + m3^4*m4^2*r99, 
          r290 = 2*m2^2 - m3^2 - m4^2, 
          r291 = 
           2*m1^4 + 2*m2^4 - m3^4 - m4^4 + m1^2*r290 - 
            m2^2*r81, r292 = m2^2 - m3^2 + m4^2, 
          r293 = 
           m1^4*r4 + m2^4*r4 - 2*m1^2*r292*r4 + 
            2*m2^2*r4*r9 - r32^2*r33^2*r99, 
          r294 = m3^2*r13 - m4^2*r131, 
          r295 = m2^2*r13 - r4*r81, 
          r296 = 
           m1^4*r131 + 2*m2^2*r294 - 2*m1^2*r295 - 
            2*m3^2*m4^2*r4 + m4^4*r8 - m2^4*r99 - 
            m3^4*r99, r297 = 2*m4^2 - m2^2*r5 + m1^2*r72, 
          r298 = 
           r107*r108*r293 - m5^2*r296 + 2*m5^4*r297, 
          r299 = -2*m5^2*r32*r33*r5 + 2*m5^4*r72, 
          r300 = 
           r130*r281 - m5^2*r289 + PP*r298 + PP^2*r299 - 
            m5^6*r264*r5 + m5^4*r291*r5, 
          r301 = 
           -2*m2^2*m3^2 - m3^2*m4^2 + m4^4 + m5^4 - 
            m5^2*r17 + m1^2*r177 + PP*r77, 
          r302 = m2^2 + 2*m3^2 - 3*m4^2, 
          r303 = -m1^2 + m2^2 - 3*m5^2, 
          r304 = 
           m1^4 - m2^2*m4^2 - m5^4 + m5^2*r149 - 
            m1^2*r302 + PP*r303, 
          r305 = 
           -((r45*TBI[d, PP, {{1, m4}, {1, m2}}]*
                 TBI[d, PP, {{2, m1}, {1, m3}}])/(4*r3)) 
+ (r76*TBI[d, PP, {{1, m3}, {1, m1}}]*
               TBI[d, PP, {{2, m2}, {1, m4}}])/(4*r3) - 
            (r80*TBI[d, PP, {{2, m1}, {1, m3}}]*
               TBI[d, PP, {{2, m2}, {1, m4}}])/2 - 
            (r45*TBI[d, PP, {{1, m4}, {1, m2}}]*
               TBI[d, PP, {{2, m3}, {1, m1}}])/(4*r3) - 
            (r84*TBI[d, PP, {{2, m2}, {1, m4}}]*
               TBI[d, PP, {{2, m3}, {1, m1}}])/2 - 
            r86*r88*TBI[d, PP, {{1, m4}, {1, m2}}]*
             TBI[d, PP, {{2, m3}, {2, m1}}] + 
            (r76*TBI[d, PP, {{1, m3}, {1, m1}}]*
               TBI[d, PP, {{2, m4}, {1, m2}}])/(4*r3) - 
            (r92*TBI[d, PP, {{2, m1}, {1, m3}}]*
               TBI[d, PP, {{2, m4}, {1, m2}}])/2 - 
            (r98*TBI[d, PP, {{2, m3}, {1, m1}}]*
               TBI[d, PP, {{2, m4}, {1, m2}}])/2 - 
            r86*r88*TBI[d, PP, {{1, m4}, {1, m2}}]*
             TBI[d, PP, {{3, m1}, {1, m3}}] - 
            r86*r88*TBI[d, PP, {{1, m4}, {1, m2}}]*
             TBI[d, PP, {{3, m3}, {1, m1}}] - 
            r3*TFI[r99, PP, 
              {{1, m1}, {1, m2}, {1, m3}, {1, m4}, 
               {1, m5}}] - 
            (r134*TVI[d, PP, 
                {{1, m5}, {1, m1}, {2, m4}, {1, m3}}])/
             (4*r3) - (r135*r136*r137*r138*
               TVI[d, PP, 
                {{1, m5}, {1, m1}, {2, m4}, {2, m3}}])/2 
- (r176*TVI[d, PP, {{1, m5}, {1, m2}, {2, m3}, 
                 {1, m4}}])/(4*r3) + 
            (r180*TVI[d, PP, 
                {{1, m5}, {1, m2}, {2, m3}, {2, m4}}])/2 
- r182*TVI[d, PP, {{1, m5}, {1, m2}, {3, m3}, 
               {1, m4}}] + 
            (r203*TVI[d, PP, 
                {{1, m5}, {1, m3}, {2, m2}, {1, m1}}])/
             (4*r3) + (r232*
               TVI[d, PP, 
                {{1, m5}, {1, m4}, {2, m1}, {1, m2}}])/
             (4*r3) + (r236*
               TVI[d, PP, 
                {{1, m5}, {1, m4}, {2, m1}, {2, m2}}])/2 
+ r240*TVI[d, PP, {{1, m5}, {1, m4}, {3, m1}, 
               {1, m2}}] + 
            (r45*TVI[d, PP, 
                {{1, m5}, {2, m1}, {1, m4}, {1, m3}}])/
             (4*r3) + (r86*r88*
               TVI[d, PP, 
                {{1, m5}, {2, m1}, {1, m4}, {2, m3}}])/2 
+ (r92*TVI[d, PP, {{1, m5}, {2, m1}, {2, m4}, 
                 {1, m3}}])/2 - 
            (r76*TVI[d, PP, 
                {{1, m5}, {2, m2}, {1, m3}, {1, m4}}])/
             (4*r3) - (r243*
               TVI[d, PP, 
                {{1, m5}, {2, m2}, {2, m3}, {1, m4}}])/2 
+ (r45*TVI[d, PP, {{1, m5}, {2, m3}, {1, m2}, 
                 {1, m1}}])/(4*r3) + 
            (r86*r88*TVI[d, PP, 
                {{1, m5}, {2, m3}, {1, m2}, {2, m1}}])/2 
+ (r246*TVI[d, PP, {{1, m5}, {2, m3}, {2, m2}, 
                 {1, m1}}])/2 - 
            (r76*TVI[d, PP, 
                {{1, m5}, {2, m4}, {1, m1}, {1, m2}}])/
             (4*r3) + (r248*
               TVI[d, PP, 
                {{1, m5}, {2, m4}, {2, m1}, {1, m2}}])/2 
+ r86*r88*TVI[d, PP, {{1, m5}, {3, m1}, {1, m4}, 
               {1, m3}}] + 
            r86*r88*TVI[d, PP, 
              {{1, m5}, {3, m3}, {1, m2}, {1, m1}}] + 
            (r275*TVI[d, PP, 
                {{2, m5}, {1, m1}, {1, m4}, {1, m3}}])/
             (4*r3) + (r276*
               TVI[d, PP, 
                {{2, m5}, {1, m1}, {1, m4}, {2, m3}}])/2 
- (r180*TVI[d, PP, {{2, m5}, {1, m1}, {2, m4}, 
                 {1, m3}}])/2 - 
            (r275*TVI[d, PP, 
                {{2, m5}, {1, m2}, {1, m3}, {1, m4}}])/
             (4*r3) + (r180*
               TVI[d, PP, 
                {{2, m5}, {1, m2}, {1, m3}, {2, m4}}])/2 
- (r276*TVI[d, PP, {{2, m5}, {1, m2}, {2, m3}, 
                 {1, m4}}])/2 - 
            (r300*TVI[d, PP, 
                {{2, m5}, {1, m3}, {1, m2}, {1, m1}}])/
             (4*r3) - (r240*
               TVI[d, PP, 
                {{2, m5}, {1, m3}, {1, m2}, {2, m1}}])/2 
- (r236*TVI[d, PP, {{2, m5}, {1, m3}, {2, m2}, 
                 {1, m1}}])/2 + 
            (r300*TVI[d, PP, 
                {{2, m5}, {1, m4}, {1, m1}, {1, m2}}])/
             (4*r3) + (r236*
               TVI[d, PP, 
                {{2, m5}, {1, m4}, {1, m1}, {2, m2}}])/2 
+ (r240*TVI[d, PP, {{2, m5}, {1, m4}, {2, m1}, 
                 {1, m2}}])/2 + 
            (r182*TVI[d, PP, 
                {{2, m5}, {2, m1}, {1, m4}, {1, m3}}])/2 
- (r301*TVI[d, PP, {{2, m5}, {2, m2}, {1, m3}, 
                 {1, m4}}])/2 - 
            (r304*TVI[d, PP, 
                {{2, m5}, {2, m3}, {1, m2}, {1, m1}}])/2 
+ (r248*TVI[d, PP, {{2, m5}, {2, m4}, {1, m1}, 
                 {1, m2}}])/2, r306}, 
         r306 = Factor /@ r305/(PP*r1*r2); r306]) /; 
         n1 === 1 && n2 === 1 && n3 === 1 && n4 === 1 && n5 === 1 &&  
       MatchQ[d, _Symbol + _Integer?Positive] && PP =!= 0 &&
       Cayley[m1, m2, m3, m4, m5, Sqrt[PP]] =!= 0];

Input (51)

general (Subscript[m, i]^2)

(Subscript[e, 51])=(Subscript[Δ, 1, 3, 4])(Subscript[ν, 1])\(1\^+\)(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))==(2(Subscript[m, 3]^2)(Subscript[ν, 3])\(3\^+\)(\(1\^-\)-\(4\^-\))+(Subscript[u, 4, 1, 3])(Subscript[ν, 1])\(1\^+\)(\(3\^-\)-\(4\^-\))+(Subscript[u, 1, 3, 4]) (d - (Subscript[ν, 1]) - 2  (Subscript[ν, 3]))+2  (Subscript[m, 3]^2) ((Subscript[ν, 1]) - (Subscript[ν, 3])))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 51]) = OperatorApplyV[(Subscript[e, 51])]

MakeFun (51)

(Subscript[s, 51])

(Subscript[h, 51]) = (Subscript[s, 51]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1

(Subscript[f, 51])=MakeFun[(Subscript[h, 51]),"(51)",IFF[(Subscript[ν, 1]) > 1 && (Subscript[Δ, 1, 3, 4]) =!= 0]]

CheckTVIRecursion[(Subscript[f, 51])]

Input (52)

general (Subscript[m, i]^2)

(Subscript[e, 52])=(Subscript[Δ, 2, 4, 6])(Subscript[ν, 2])\(2\^+\)(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))==(2(Subscript[m, 1]^2)(Subscript[ν, 1])\(1\^+\)(\(4\^-\)-\(2\^-\))+2(Subscript[m, 3]^2)(Subscript[ν, 3])\(3\^+\)(\(4\^-\)-\(2\^-\))-2(Subscript[m, 4]^2)(Subscript[ν, 4])\(4\^+\)\(2\^-\)+(2  (Subscript[m, 2]^2) - (Subscript[u, 4, 2, 6]))(Subscript[ν, 2])\(2\^+\)\(4\^-\)-((Subscript[Σ, 4]) + 2)(\(4\^-\)-\(2\^-\))+(Subscript[u, 2, 4, 6]) (d - 3  (Subscript[ν, 2]))+2  (Subscript[m, 4]^2) ((Subscript[ν, 4]) - (Subscript[ν, 2])))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 52]) = OperatorApplyV[(Subscript[e, 52])]

MakeFun (52)

(Subscript[h, 52]) = νexplicit[
          (Subscript[s, 52])] /. (Subscript[ν, 2]) -> (Subscript[ν, 2]) - 1

(Subscript[f, 52])=MakeFun[(Subscript[h, 52]),"(52)",IFF[(Subscript[ν, 2]) > 1 && (Subscript[Δ, 2, 4, 6]) =!= 0]]

CheckTVIRecursion[(Subscript[f, 52])]

MakeFun (53)

(Subscript[h, 53]) = 
      Interchange[(Subscript[s, 51]), 
          1 ↔ 3] /. (Subscript[ν, 3]) -> (Subscript[ν, 3]) - 1

(Subscript[f, 53])=MakeFun[(Subscript[h, 53]),"(53)",IFF[(Subscript[ν, 3]) > 1 && (Subscript[Δ, 3, 1, 4]) =!= 0]]

CheckTVIRecursion[(Subscript[f, 53])]

Input (54)

general (Subscript[m, i]^2)

(Subscript[e, 54])=2(Subscript[m, 3]^2)(Subscript[ν, 3])\(3\^+\)(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))==((Subscript[u, 4, 1, 3])(Subscript[ν, 1])\(1\^+\)+(Subscript[ν, 1])\(1\^+\)(\(4\^-\)-\(3\^-\))+
				(d - (Subscript[ν, 1]) - 2  (Subscript[ν, 3])))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 54]) = OperatorApplyV[(Subscript[e, 54])]

MakeFun (54)

(Subscript[h, 54]) = (Subscript[s, 54]) /. (Subscript[ν, 3]) -> (Subscript[ν, 3]) - 1

(Subscript[f, 54])=MakeFun[(Subscript[h, 54]),"(54)",IFF[ (Subscript[ν, 3]) > 1 &&  m3 =!= 0]]

CheckTVIRecursion[(Subscript[f, 54])]

Input (55)

general (Subscript[m, i]^2)

(Subscript[e, 55])=2 (Subscript[m, 4]^2)(Subscript[ν, 4])\(4\^+\)(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))==((Subscript[u, 3, 1, 4])(Subscript[ν, 1])\(1\^+\)+(Subscript[u, 6, 2, 4])(Subscript[ν, 2])\(2\^+\)+
				(Subscript[ν, 1])\(1\^+\)(\(3\^-\)-\(4\^-\))-(Subscript[ν, 2])\(2\^+\)\(4\^-\)+(d - (Subscript[ν, 1]) - (Subscript[ν, 2]) - 2  (Subscript[ν, 4])))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 55]) = OperatorApplyV[(Subscript[e, 55])]

MakeFun (55)

(Subscript[h, 55]) = (Subscript[s, 55]) /. (Subscript[ν, 4]) -> (Subscript[ν, 4]) - 1

(Subscript[f, 55])=MakeFun[(Subscript[h, 55]),"(55)",IFF[(Subscript[ν, 4]) > 1 &&  m4 =!= 0  &&  (Subscript[ν, 3]) > 1]]

CheckTVIRecursion[(Subscript[f, 55])]

Input (56)

general (Subscript[m, i]^2)

(Subscript[e, 56])=2 (Subscript[m, 4]^2)(Subscript[ν, 4])\(4\^+\)(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))==(-2(Subscript[m, 1]^2)(Subscript[ν, 1])\(1\^+\)+(Subscript[u, 6, 2, 4])(Subscript[ν, 2])\(2\^+\)-2(Subscript[m, 3]^2)(Subscript[ν, 3])\(3\^+\)-
				(Subscript[ν, 2])\(2\^+\)\(4\^-\)+(2  d - 2  (Subscript[ν, 1]) - (Subscript[ν, 2]) - 2  (Subscript[ν, 3]) - 
                    2  (Subscript[ν, 4])))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 56]) = OperatorApplyV[(Subscript[e, 56])]

MakeFun (56)

(Subscript[h, 56]) = (Subscript[s, 56]) /. (Subscript[ν, 4]) -> (Subscript[ν, 4]) - 1

(Subscript[f, 56])=MakeFun[(Subscript[h, 56]),"(56)",IFF[(Subscript[ν, 4]) > 1 &&  m4 =!= 0  &&  (Subscript[ν, 3]) === 1]]

CheckTVIRecursion[(Subscript[f, 56])]

Input (59)

(Subscript[Δ, 1, 3, 4])== 0 but all  (Subscript[m, i]^2) != 0

(Subscript[e, 59])=\((2 \( m\_1\%2\) \(u\_\(1, 3, 4\)\) \((d - ν\_1 - 
                      2  ν\_3)\) + \((ν\_3 - 1)\) 
                  u\_\(4, 1, 3\)\%2 + 
	
                4 \( m\_1\%2\) \(m\_3\%2\) \((ν\_1 - ν\_3)\))\)(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))==((Subscript[u, 4, 1, 3])(Subscript[ν, 3])\(3\^+\)\(4\^-\)(\(4\^-\)-\(1\^-\))-
	4(Subscript[m, 1]^2)(Subscript[m, 3]^2)(Subscript[ν, 3])\(3\^+\)\(1\^-\)+(4  (Subscript[m, 1]^2) (Subscript[m, 3]^2) + 
                      (Subscript[u, 4, 1, 3]^2))(Subscript[ν, 3])\(3\^+\)\(4\^-\)+(Subscript[u, 4, 1, 3])((Subscript[ν, 3]) - 1)\(1\^-\)-
		(d - 2  (Subscript[ν, 1]) - (Subscript[ν, 3]) + 1)(Subscript[u, 4, 1, 
                      3])\(3\^-\)+(d - 2  (Subscript[ν, 1]) - 2  (Subscript[ν, 3]) + 1)(Subscript[u, 4, 1, 
                      3])\(4\^-\))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 59]) = OperatorApplyV[(Subscript[e, 59])]

MakeFun (59)

(Subscript[h, 59]) = (Subscript[s, 59])

(Subscript[f, 59])=MakeFun[(Subscript[h, 59]),"(59)",IFF[(Subscript[Δ, 1, 3, 4]) == 
                0 && (4 (Subscript[m, 3]^2) (Subscript[m, 1]^2) - 4 (Subscript[m, 3]^2) (Subscript[ν, 3]) (Subscript[m, 1]^2) + 
                    2 d (Subscript[u, 1, 3, 4]) (Subscript[m, 1]^2) - 
                    4 (Subscript[ν, 3]) (Subscript[u, 1, 3, 4]) (Subscript[m, 1]^2) - 
                    2 (Subscript[u, 1, 3, 4]) (Subscript[m, 1]^2) + (Subscript[ν, 3]) (Subscript[u, 4, 1, 3]^2) - (Subscript[u, 4, 1, 3]^2))   =!= 0 && n1  ===  1 &&  n2  ===  1 &&  
              n4  ===  1 && MatchQ[d, _Integer]]]

CheckTVIRecursion not applicable .

Input (60)

general (Subscript[m, i]^2)

(Subscript[e, 60])=2(Subscript[m, 1]^2)(Subscript[ν, 1])\(1\^+\)(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))==((Subscript[u, 4, 1, 3])(Subscript[ν, 3])\(3\^+\)+(Subscript[ν, 3])\(3\^+\)(\(4\^-\)-\(1\^-\))+
				(d - (Subscript[ν, 3]) - 2  (Subscript[ν, 1])))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 60]) = OperatorApplyV[(Subscript[e, 60])]

MakeFun (60)

(Subscript[h, 60]) = (Subscript[s, 60]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1

(Subscript[notnecessaryf, 60])=MakeFun[(Subscript[h, 60]),"(60)",IFF[(Subscript[ν, 1]) > 1 &&  m1 =!= 0]]

CheckTVIRecursion[(Subscript[notnecessaryf, 60])]

Input (61)

( (Subscript[m, 1]^2)== 0 && (Subscript[m, 4]^2)== (Subscript[m, 3]^2) )  ⟹ ( (Subscript[Δ, 1, 3, 4])== 0 )

(Subscript[e, 61])=(d - 2  (Subscript[ν, 1]) - (Subscript[ν, 3]))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))==(Subscript[ν, 3])\(3\^+\)(\(1\^-\)-\(4\^-\))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 61]) = OperatorApplyV[(Subscript[e, 61])]

For  ( (Subscript[m, 3]^2)== 0 && (Subscript[m, 4]^2)== (Subscript[m, 1]^2) )  ⟹ ( (Subscript[Δ, 1, 3, 4])== 0 )
use ((Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))==(Subscript[V, (Subscript[ν, 3])(Subscript[ν, 2])(Subscript[ν, 1])(Subscript[ν, 4])]^((d))))  and switch the masses accordingly.

MakeFun (61)

( (Subscript[m, 1]^2)== 0 && (Subscript[m, 4]^2)== (Subscript[m, 3]^2) )  ⟹ ( (Subscript[Δ, 1, 3, 4])== 0 )

(Subscript[h, 61]) = (Subscript[s, 61])

(Subscript[f, 61])=MakeFun[(Subscript[h, 61]),"(61)",IFF[(Subscript[m, 1])===0&&(Subscript[m, 3])===(Subscript[m, 4])],{n1 :> 1, n4 :> 1}]

CheckTVIRecursion not applicable .

MakeFun (6153)

( (Subscript[m, 3]^2)== 0 && (Subscript[m, 4]^2)== (Subscript[m, 1]^2))  ⟹ ( (Subscript[Δ, 1, 3, 4])== 0 )

(Subscript[h, 6153]) = Interchange[(Subscript[s, 61]), 1 ↔ 3]

(Subscript[f, 6153])=MakeFun[(Subscript[h, 6153]),"(6153)",IFF[(Subscript[m, 3])===0&&(Subscript[m, 1])===(Subscript[m, 4])],{n3 :> 1, n4 :> 1}]

CheckTVIRecursion not applicable .

Input (62)

( (Subscript[m, 4]^2)== 0 && (Subscript[m, 1]^2)== (Subscript[m, 3]^2) ) ⟹ ( (Subscript[Δ, 1, 3, 4])== 0 )
lower (Subscript[ν, 2]) via (52) to one first

(Subscript[e, 62])=((p^2)-(Subscript[m, 2]^2))((Subscript[ν, 1]) - 2  (Subscript[ν, 2]) + 2  (Subscript[ν, 4]) + 2)\(4\^+\)(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))==
		(((Subscript[m, 1]^2) - (Subscript[u, 6, 1, 2]))(Subscript[ν, 1])\(1\^+\)+4(Subscript[m, 2]^2)(Subscript[ν, 2])\(2\^+\)+2(Subscript[m, 1]^2)(Subscript[ν, 3])\(3\^+\)+
				((Subscript[Σ, 4])-2(Subscript[m, 1]^2)(Subscript[ν, 1])\(1\^+\)-2(Subscript[m, 1]^2)(Subscript[ν, 3])\(3\^+\))\(2\^-\)\(4\^+\)+((p^2)-(Subscript[m, 2]^2))(Subscript[ν, 1])\(1\^+\)\(4\^+\)\(3\^-\)-
				(Subscript[Σ, 4]))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 62]) = OperatorApplyV[(Subscript[e, 62])]

MakeFun (62)

(Subscript[h, 62]) = νexplicit[
          (Subscript[s, 62])] /. (Subscript[ν, 4]) -> (Subscript[ν, 4]) - 1

(Subscript[f, 62])=MakeFun[(Subscript[h, 62]),"(62)",IFF[(p^2)=!=(Subscript[m, 2]^2)&&(Subscript[ν, 1]) - 2 (Subscript[ν, 2]) + 2 (Subscript[ν, 4]) =!= 0],{(Subscript[m, 4]):>0,(Subscript[m, 3]):>(Subscript[m, 1]),n2 :> 1}]

CheckTVIRecursion not applicable .

Input (63)

(Subscript[m, 4]^2)== 0  &&  (Subscript[Δ, 1, 3, 4]) != 0
replace (55) by the following

(Subscript[e, 63])=((p^2)-(Subscript[m, 2]^2))(((Subscript[m, 3]^2) - (Subscript[m, 1]^2)) (d - 2  (Subscript[ν, 2]) - 
                      2  (Subscript[ν, 3]) + 2  (Subscript[ν, 4])) + 
                2  (Subscript[m, 3]^2) ((Subscript[ν, 3]) - (Subscript[ν, 1])))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))==
		(((Subscript[m, 1]^2) - (Subscript[m, 3]^2))((2(Subscript[m, 1]^2)(Subscript[ν, 1])\(1\^+\)+2(Subscript[m, 3]^2)(Subscript[ν, 3])\(3\^+\)-(Subscript[Σ, 4])-2)(\(2\^-\)-\(4\^-\))-4(Subscript[m, 2]^2)(Subscript[ν, 2])\(2\^+\)\(4\^-\))+
				2((p^2)-(Subscript[m, 2]^2))((Subscript[m, 3]^2)(Subscript[ν, 3])\(3\^+\)(\(1\^-\)-\(4\^-\))+(Subscript[m, 1]^2)(Subscript[ν, 1])\(1\^+\)(\(4\^-\)-\(3\^-\))))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 63]) = OperatorApplyV[(Subscript[e, 63])]

MakeFun (63)

(Subscript[m, 4]^2)== 0  &&  (Subscript[Δ, 1, 3, 4]) != 0

(Subscript[h, 63]) = νexplicit[(Subscript[s, 63])]

(Subscript[f, 63])=MakeFun[(Subscript[h, 63]),"(63)",IFF[PP  =!=  m2^2 && Cayley[1, 3, 4][{m1, m2, m3, m4}] =!= 
                0],{(Subscript[m, 4]):>0}]

CheckTVIRecursion[(Subscript[f, 63])]

(Subscript[f, 63])=MakeFun[(Subscript[h, 63]),"(63)",IFF[PP  =!=  m2^2 && Cayley[1, 3, 4][{m1, m2, m3, m4}] =!= 
                0],{(Subscript[m, 4]):>0,n1 :> 1,n3 :> 1}]

CheckTVIRecursion not applicable for special indices since A's and 
B's appear .

MakeFun (6552)

(Subscript[r, 6552]) = νexplicit[(Subscript[e, 52])] /. (Subscript[Δ, 2, 4, 6]) -> 
          0

formal=Solve[(Subscript[e, 55]) /. TVI[__] -> 1,\(4\^+\)][[1, 1]]/.(Subscript[ν, 2]) -> (Subscript[ν, 2]) - 1

(Subscript[r, 6552]) /. formal

(Subscript[n, 6552]) = OperatorApplyV[(Subscript[r, 6552]) /. formal]

(Subscript[n, 6552])[[2, 4]]

(Subscript[s, 6552]) = -(Subscript[n, 6552])[[2, 
            4]] == ((Subscript[n, 6552])[[2]] - 
            (Subscript[n, 6552])[[2, 4]])

(Subscript[f, 6552])=MakeFun[(Subscript[s, 6552]),"(6552)",IFF[(Subscript[Δ, 2, 4, 6]) === 0&&Expand[2 ((Subscript[ν, 2]) - (Subscript[ν, 4])) (Subscript[m, 4]^2)+(3 (Subscript[ν, 2]) - d) (Subscript[u, 2, 4, 6])+(Subscript[ν, 2]) (Subscript[u, 6, 2, 4])]=!=0]]

CheckTVIRecursion not applicable .

MakeFun (6552)

(Subscript[r, 6552]) = νexplicit[(Subscript[e, 52])] /. (Subscript[Δ, 2, 4, 6]) -> 
          0

(Subscript[n, 6552]) = OperatorApplyV[(Subscript[r, 6552])]

(Subscript[n, 6552])[[2, 5]]

(Subscript[s, 6552]) = -(Subscript[n, 6552])[[2, 
            5]] == ((Subscript[n, 6552])[[2]] - 
            (Subscript[n, 6552])[[2, 5]])

(Subscript[f, 6552])=MakeFun[(Subscript[s, 6552]),"(6552)",IFF[(Subscript[Δ, 2, 4, 6]) === 0 && 
              Expand[(-2 (Subscript[ν, 2]) (Subscript[m, 4]^2) + 2 (Subscript[ν, 4]) (Subscript[m, 4]^2) + 
                      d (Subscript[u, 2, 4, 6]) - 3 (Subscript[ν, 2]) (Subscript[u, 2, 4, 6]))] =!= 
                0]]

CheckTVIRecursion not applicable .

MakeFun (6553)

(Subscript[r, 6553]) = νexplicit[(Subscript[s, 55])] /. (Subscript[u, 3, 1, 4]) -> 0 /. 
          (Subscript[u, 6, 2, 4]) -> 0 /. (Subscript[m, 4]) -> 0

(Subscript[h, 6553]) = -(Subscript[r, 6553])[[2, 
            1]] == ((Subscript[r, 6553])[[2]] - 
            (Subscript[r, 6553])[[2, 1]])

(Subscript[f, 6553])=MakeFun[(Subscript[h, 6553]),"(6553)",IFF[(Subscript[u, 6, 2, 4]) === 0 && (Subscript[u, 1, 3, 4]) === 0],m4 :> 0]

CheckTVIRecursion not applicable .

Input (66)

general (Subscript[m, i]^2)

(Subscript[e, 66])=4(p^2)(Subscript[ν, 1])\(1\^+\)(Subscript[ν, 2])\(2\^+\)(Subscript[ν, 3])\(3\^+\)(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d + 2)))==-((6  d-6  (Subscript[ν, 1])-5  (Subscript[ν, 2])-6  (Subscript[ν, 3])-4  (Subscript[ν, 4])-6(Subscript[m, 1]^2)(Subscript[ν, 1])\(1\^+\))+((Subscript[u, 6, 2, 4]) - 4  (Subscript[m, 2]^2))(Subscript[ν, 2])\(2\^+\)-6(Subscript[m, 3]^2)(Subscript[ν, 3])\(3\^+\)+2((Subscript[u, 6, 2, 4]) - (Subscript[m, 4]^2))(Subscript[ν, 4])\(4\^+\)-(Subscript[ν, 2])\(2\^+\)\(4\^-\)-2(Subscript[ν, 4])\(4\^+\)\(2\^-\))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 66]) = OperatorApplyV[(Subscript[e, 66])]

MakeFun (66)

(Subscript[h, 66]) = 
      (Subscript[s, 66]) /. {(Subscript[ν, 1]) :> (Subscript[ν, 1]) - 1, (Subscript[ν, 2]) :> (Subscript[ν, 2]) - 
              1, (Subscript[ν, 3]) :> (Subscript[ν, 3]) - 1, d :> d - 2}

(Subscript[f, 66])=MakeFun[(Subscript[h, 66]),"(66)",IFF[PP =!= 0 && (Subscript[ν, 1]) > 1 && (Subscript[ν, 2]) > 1 && (Subscript[ν, 3]) > 1 && 
              MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTVIRecursion[(Subscript[f, 66])]

Input (alternative67)

we dereived an alternative formula for Tarasov's (67), but do not save it by default.

general (Subscript[m, i]^2)

(Subscript[e, 67])=4(p^2)(Subscript[ν, 1])((Subscript[ν, 1]) + 1)\(1\^+\)\(1\^+\)(Subscript[ν, 2])\(2\^+\)(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d + 2))) ==((6  d - 6  (Subscript[ν, 1]) - 5  (Subscript[ν, 2]) - 6  (Subscript[ν, 3]) - 
                    4  (Subscript[ν, 4]))-2((Subscript[u, 2, 4, 6]) + 3  (Subscript[m, 1]^2))(Subscript[ν, 1])\(1\^+\)+((Subscript[u, 6, 2, 4]) - 4  (Subscript[m, 2]^2))(Subscript[ν, 2])\(2\^+\)-6(Subscript[m, 3]^2)(Subscript[ν, 3])\(3\^+\)+2((Subscript[u, 6, 2, 4]) - (Subscript[m, 4]^2))(Subscript[ν, 4])\(4\^+\)+2(Subscript[ν, 1])\(1\^+\)(\(4\^-\)-\(2\^-\))-(Subscript[ν, 2])\(2\^+\)\(4\^-\)-2(Subscript[ν, 4])\(4\^+\)\(2\^-\))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 67]) = OperatorApplyV[(Subscript[e, 67])]

Input (67)

general (Subscript[m, i]^2)

(Subscript[e, 67])=4(p^2)(Subscript[ν, 1])((Subscript[ν, 1]) + 1)\(1\^+\)\(1\^+\)(Subscript[ν, 2])\(2\^+\)(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d + 2))) ==(2 (2  d - 2  (Subscript[ν, 1]) - 2  (Subscript[ν, 2]) - 
                      2  (Subscript[ν, 3]) - (Subscript[ν, 4]))-2((Subscript[u, 2, 4, 6]) + 2  (Subscript[m, 1]^2))(Subscript[ν, 1])\(1\^+\)+-4(Subscript[m, 2]^2)(Subscript[ν, 2])\(2\^+\)-4(Subscript[m, 3]^2)(Subscript[ν, 3])\(3\^+\)+2(Subscript[u, 6, 2, 4])(Subscript[ν, 4])\(4\^+\)+2(Subscript[ν, 1])\(1\^+\)(\(4\^-\)-\(2\^-\))-2(Subscript[ν, 4])\(4\^+\)\(2\^-\))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

-4 (Subscript[ν, 2]) (Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d))) (Subscript[m, 2]^2)-2 (Subscript[ν, 4]) (Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))+2 (2 d - 2 (Subscript[ν, 1]) - 2 (Subscript[ν, 2]) - 
              2 (Subscript[ν, 3]) - (Subscript[ν, 4])) (Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))-2 ((Subscript[m, 2]^2) + (Subscript[m, 4]^2) - (p^2)) (Subscript[ν, 4]) (Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))-4 (Subscript[m, 3]^2) (Subscript[ν, 3]) (Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))-2 (Subscript[ν, 1]) (Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))+2 (Subscript[ν, 1]) (Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))-2 (2 (Subscript[m, 1]^2) + (Subscript[m, 2]^2) - (Subscript[m, 4]^2) - (p^2)) (Subscript[ν, 1]) (Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)))-4 (p^2) (Subscript[ν, 1]) ((Subscript[ν, 1]) + 1) (Subscript[ν, 2]) (Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d + 2)))

(Subscript[s, 67]) = OperatorApplyV[(Subscript[e, 67])]

MakeFun (67)

(Subscript[h, 67]) = 
      (Subscript[s, 67]) /. {(Subscript[ν, 1]) :> (Subscript[ν, 1]) - 2, (Subscript[ν, 2]) :> (Subscript[ν, 2]) - 1, 
          d :> d - 2}

(Subscript[f, 67])=MakeFun[(Subscript[h, 67]),"(67)",IFF[PP =!= 0 && (Subscript[ν, 1]) > 2 && (Subscript[ν, 2]) > 1 && 
              MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTVIRecursion[(Subscript[f, 67])]

MakeFun (6713)

(Subscript[h, 6713]) = 
      Interchange[(Subscript[s, 67]), 
          1 ↔ 
            3] /. {(Subscript[ν, 3]) :> (Subscript[ν, 3]) - 2, (Subscript[ν, 2]) :> (Subscript[ν, 2]) - 1, 
          d :> d - 2}

(Subscript[f, 6713])=MakeFun[(Subscript[h, 6713]),"(6713)",IFF[PP =!= 0 && (Subscript[ν, 3]) > 2 && (Subscript[ν, 2]) > 1 && 
              MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTVIRecursion[(Subscript[f, 6713])]

Input (68)

general (Subscript[m, i]^2)

(Subscript[e, 68])=4(p^2)(Subscript[m, 4]^2)(Subscript[ν, 1])\(1\^+\)(Subscript[ν, 2])((Subscript[ν, 2]) + 1)\(2\^+\)\(2\^+\)(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d + 2)))==-((((Subscript[m, 1]^2) - (Subscript[m, 3]^2)) (2  d - 
                            2  (Subscript[ν, 1]) - (Subscript[ν, 2]) - 2  (Subscript[ν, 3]) - 
                            2  (Subscript[ν, 4])) + (Subscript[m, 4]^2) (2  (Subscript[ν, 1]) + (Subscript[ν, 2]) - 2  (Subscript[ν, 3])))-(Subscript[u, 1, 3, 4])(2(Subscript[m, 1]^2)(Subscript[ν, 1])\(1\^+\)+(Subscript[u, 2, 4, 6])(Subscript[ν, 2])\(2\^+\))+(Subscript[u, 3, 1, 4])(2(Subscript[m, 3]^2)(Subscript[ν, 3])\(3\^+\)+(Subscript[ν, 2])\(2\^+\)\(4\^-\))-2(Subscript[m, 4]^2)((Subscript[ν, 2])\(2\^+\)+(Subscript[ν, 4])\(4\^+\))(\(3\^-\)-\(1\^-\)))(Subscript[V, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])(Subscript[ν, 4])]^((d)));

(Subscript[s, 68]) = OperatorApplyV[(Subscript[e, 68])]

MakeFun (68)

(Subscript[h, 68]) = 
      (Subscript[s, 68]) /. {(Subscript[ν, 1]) :> (Subscript[ν, 1]) - 1, (Subscript[ν, 2]) :> (Subscript[ν, 2]) - 2, 
          d :> d - 2}

(Subscript[f, 68])=MakeFun[(Subscript[h, 68]),"(68)",IFF[PP =!= 0 && m4 =!= 0 && (Subscript[ν, 1]) > 1 && (Subscript[ν, 2]) > 2 && 
              MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTVIRecursion[(Subscript[f, 68])]

MakeFun (6813)

(Subscript[h, 6813]) = 
      Interchange[(Subscript[s, 68]), 
          1 ↔ 
            3] /. {(Subscript[ν, 2]) :> (Subscript[ν, 2]) - 2, (Subscript[ν, 3]) :> (Subscript[ν, 3]) - 1, 
          d :> d - 2}

(Subscript[f, 6813])=MakeFun[(Subscript[h, 6813]),"(6813)",IFF[PP =!= 0 && m4 =!= 0 && (Subscript[ν, 2]) > 2 && (Subscript[ν, 3]) > 1 && 
              MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTVIRecursion[(Subscript[f, 6813])]

Input (70)

general (Subscript[m, i]^2)

(Subscript[e, 70])=(3  d - 10)(((d - 3))^2)(Subscript[m, 4]^2)(Subscript[V, 1111]^((d)))==(1/(4(p^2)))((3  d - 10)(Subscript[Δ, 1, 3, 4])((Subscript[Δ, 2, 4, 6])+(Subscript[u, 2, 4, 6])(\(2\^-\)-\(4\^-\)))-8(((d - 4))^2)(p^2)(Subscript[m, 3]^2)(Subscript[m, 4]^2)\(4\^-\))(Subscript[V, 1111]^((d - 2)))+
			((d - 3)(Subscript[m, 4]^2)(4(Subscript[m, 1]^2)(d - 4)\(1\^+\)\(1\^+\)-8(d - 3)(Subscript[m, 2]^2)\(2\^+\)\(2\^+\)+(3  d - 10)(d - 3)\(2\^+\))+(Subscript[u, 6, 2, 4])((1/2)(3  d - 10)(Subscript[u, 1, 3, 4])(\(1\^+\)-\(3\^+\))+(d - 2)(d - 3)(Subscript[m, 4]^2)\(1\^+\))\(2\^+\)-(Subscript[m, 4]^2)((((d - 4))^2)(Subscript[u, 1, 3, 4])\(2\^+\)+(3  d - 10)(Subscript[u, 4, 1, 3])\(3\^+\))\(1\^+\))\(4\^-\)(Subscript[V, 1111]^((d)))-(d - 3)(Subscript[m, 4]^2)(4(d - 4)(Subscript[m, 1]^2)\(1\^+\)+(d - 2)((Subscript[u, 6, 2, 4])\(2\^+\)-2(Subscript[m, 4]^2)\(4\^+\)))\(1\^+\)\(3\^-\)(Subscript[V, 1111]^((d)))+(1/2)(3  d - 10)(Subscript[u, 1, 3, 4])((Subscript[u, 6, 2, 4])\(2\^+\)-2(Subscript[m, 4]^2)\(4\^+\))(\(3\^+\)\(1\^-\)-\(1\^+\)\(3\^-\))(Subscript[V, 1111]^((d)));

(Subscript[s, 70]) = OperatorApplyV[(Subscript[e, 70])]

MakeFun (70)

(Subscript[h, 70]) = (Subscript[s, 70])

(Subscript[f, 70])=MakeFun[(Subscript[h, 70]),"(70)",IFF[m4 =!= 0  &&  PP =!= 0 && 
              MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTVIRecursion not applicable for special indices since A's and 
B's appear .

Input (71)

general (Subscript[m, i]^2) and (Subscript[ν, 1]), (Subscript[ν, 2]) > 0

(Subscript[e, 71])=2(Subscript[D, 1, 2, 3])(Subscript[ν, 1])\(1\^+\)(Subscript[ν, 2])\(2\^+\)(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==(2(Subscript[h, 1, 2, 3])(Subscript[ν, 1])\(1\^+\)+2(Subscript[h, 2, 1, 3])(Subscript[ν, 2])\(2\^+\)+4(Subscript[m, 3]^2)(Subscript[σ, 1, 2, 3])(Subscript[ν, 3])\(3\^+\)+(Subscript[m, 3]^2)(Subscript[ϕ, 2, 1, 3])(Subscript[ν, 2])\(2\^+\)(Subscript[ν, 3])\(3\^+\)\(1\^-\)+(Subscript[m, 3]^2)(Subscript[ϕ, 1, 2, 3])(Subscript[ν, 1])\(1\^+\)(Subscript[ν, 3])\(3\^+\)\(2\^-\)-2(Subscript[ρ, 1, 2, 3])(Subscript[ν, 1])\(1\^+\)(Subscript[ν, 2])\(2\^+\)\(3\^-\)+(1/2) (Subscript[Σ, 3]) (d - (Subscript[ν, 1]) - (Subscript[ν, 2]) - (Subscript[ν, 3])) (Subscript[ϕ, 3, 2, 1]))(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)));

(Subscript[s, 71]) = OperatorApplyJ[(Subscript[e, 71])]

MakeFun (71)

(Subscript[h, 71]) = νexplicit[
            (Subscript[s, 71])] /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1 /. (Subscript[ν, 2]) -> (Subscript[ν, 2]) - 
            1

(Subscript[f, 71]) = 
      MakeFun[(Subscript[h, 71]), "(71)", 
        IFF[(Subscript[ν, 1]) > 1 && (Subscript[ν, 2]) > 1 && (Subscript[D, 1, 2, 3]) =!= 0]]

CheckTJIRecursion[(Subscript[f, 71])]

MakeFun (7123)

(Subscript[h, 7123]) = Interchange[(Subscript[s, 71]), 
            2 ↔ 3] /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 
              1 /. (Subscript[ν, 3]) -> (Subscript[ν, 3]) - 1

(Subscript[f, 7123]) = 
      MakeFun[(Subscript[h, 7123]), "(7123)", 
        IFF[(Subscript[ν, 1]) > 1 && (Subscript[ν, 3]) > 1 && (Subscript[D, 1, 3, 2]) =!= 0]]

CheckTJIRecursion[(Subscript[f, 7123])]

MakeFun (7113)

(Subscript[h, 7113]) = Interchange[(Subscript[s, 71]), 
            1 ↔ 3] /. (Subscript[ν, 2]) -> (Subscript[ν, 2]) - 
              1 /. (Subscript[ν, 3]) -> (Subscript[ν, 3]) - 1

(Subscript[f, 7113]) = 
      MakeFun[(Subscript[h, 7113]), "(7113)", 
        IFF[(Subscript[ν, 2]) > 1 && (Subscript[ν, 3]) > 1 && (Subscript[D, 3, 2, 1]) =!= 0]]

CheckTJIRecursion[(Subscript[f, 7113])]

Input (78)

general (Subscript[m, i]^2) and (Subscript[ν, 1]) > 0

(Subscript[e, 78])=2(Subscript[m, 1]^2) (Subscript[D, 1, 2, 3])(Subscript[ν, 1])((Subscript[ν, 1]) + 1)\(1\^+\)\(1\^+\)(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==(-(Subscript[Σ, 3]) (d - (Subscript[ν, 1]) - (Subscript[ν, 2]) - (Subscript[ν, 3])) (Subscript[ρ, 1, 2, 3])+(Subscript[m, 2]^2)(Subscript[m, 3]^2)(Subscript[ϕ, 1, 2, 
                      3])(Subscript[ν, 2])\(2\^+\)(Subscript[ν, 3])\(3\^+\)\(1\^-\)+(Subscript[m, 1]^2)(Subscript[m, 3]^2)(Subscript[ϕ, 2, 1, 
                      3])(Subscript[ν, 1])\(1\^+\)(Subscript[ν, 3])\(3\^+\)\(2\^-\)+(Subscript[m, 1]^2)(Subscript[m, 2]^2)(Subscript[ϕ, 3, 1, 
                      2])(Subscript[ν, 1])\(1\^+\)(Subscript[ν, 2])\(2\^+\)\(3\^-\)+(d - 2 - 2  (Subscript[ν, 1]))(Subscript[D, 1, 2, 
                      3])(Subscript[ν, 1])\(1\^+\)+(Subscript[m, 1]^2)(Subscript[S, 1, 2, 3])(Subscript[ν, 1])\(1\^+\)+(Subscript[m, 2]^2)(Subscript[S, 2, 1, 3])(Subscript[ν, 2])\(2\^+\)+(Subscript[m, 3]^2)(Subscript[S, 3, 1, 2])(Subscript[ν, 3])\(3\^+\))(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)));

(Subscript[s, 78]) = OperatorApplyJ[(Subscript[e, 78])]

MakeFun (78)

(Subscript[h, 78]) = νexplicit[(Subscript[s, 78])] /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 2

(Subscript[f, 78]) = 
      MakeFun[(Subscript[h, 78]), "(78)", 
        IFF[(Subscript[ν, 1]) > 2 && m1 =!= 0 && (Subscript[D, 1, 2, 3]) =!= 0]]

CheckTJIRecursion[(Subscript[f, 78])]

Input (80)

(Subscript[m, 1]^2) == 0 and (Subscript[ν, 1]) > 0

(Subscript[e, 80])=(Subscript[ν, 1]) (d - 2 (Subscript[ν, 1]) - 2)(Subscript[Δ, 2, 3, 6])\(1\^+\)(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==(-4(Subscript[m, 2]^2)(Subscript[m, 3]^2)(Subscript[ν, 2])\(2\^+\)(Subscript[ν, 3])\(3\^+\)\(1\^-\)+
				2(Subscript[m, 2]^2)(((p^2)-(Subscript[m, 2]^2))(2  d - 2  (Subscript[ν, 1]) - (Subscript[ν, 2]) - 
                            2  (Subscript[ν, 3]) - 1)-(Subscript[m, 3]^2) (2  (Subscript[ν, 1]) - (Subscript[ν, 2]) - 1))(Subscript[ν, 2])\(2\^+\)+
			  2 (Subscript[m, 3]^2)(((p^2)-(Subscript[m, 3]^2))(2  d - 2  (Subscript[ν, 1]) - 
                            2  (Subscript[ν, 2]) - (Subscript[ν, 3]) - 1)-(Subscript[m, 2]^2) (2  (Subscript[ν, 1]) - (Subscript[ν, 3]) - 1))(Subscript[ν, 3])\(3\^+\)-
				(Subscript[Σ, 3]) (d - (Subscript[ν, 1]) - (Subscript[ν, 2]) - (Subscript[ν, 3])) (Subscript[u, 6, 2, 3]))(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)));

(Subscript[s, 80]) = OperatorApplyJ[(Subscript[e, 80])]

MakeFun (80)

(Subscript[h, 80]) = νexplicit[
          (Subscript[s, 80])] /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1

(Subscript[f, 80]) = 
      MakeFun[(Subscript[h, 80]), "(80)", 
        IFF[(Subscript[ν, 1]) > 1 && (Subscript[Δ, 2, 3, 6]) =!= 0], 
        m1 :> 0]

CheckTJIRecursion[(Subscript[f, 80])]

Input (81)

general (Subscript[m, i]^2)and (Subscript[ν, 1]), (Subscript[ν, 2]) > 0

(Subscript[e, 81])=(p^2)((Subscript[Σ, 3]) + 2)(Subscript[ν, 1])\(1\^+\)(Subscript[ν, 2])\(2\^+\)(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d + 2)))==((p^2)(d - 2  (Subscript[ν, 3]))+(Subscript[m, 1]^2) (d - (Subscript[ν, 1]) - 2  (Subscript[ν, 2]))+(Subscript[m, 2]^2) (d - 2  (Subscript[ν, 1]) - (Subscript[ν, 2]))-2  
                      (Subscript[m, 3]^2) (d - (Subscript[ν, 1]) - (Subscript[ν, 2]) - (Subscript[ν, 3]))-2(Subscript[m, 3]^2)((p^2)-(Subscript[m, 3]^2))(Subscript[ν, 3])\(3\^+\)+(Subscript[m, 1]^2)((p^2)-(Subscript[m, 1]^2)-3  (Subscript[m, 2]^2)+3  (Subscript[m, 3]^2)-\(2\^-\)+\(3\^-\))(Subscript[ν, 1])\(1\^+\)+(Subscript[m, 2]^2)((p^2)-3  (Subscript[m, 1]^2)-(Subscript[m, 2]^2)+3  (Subscript[m, 3]^2)-\(1\^-\)+\(3\^-\))(Subscript[ν, 2])\(2\^+\))(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)));

(Subscript[s, 81]) = OperatorApplyJ[(Subscript[e, 81])]

MakeFun (81)

(Subscript[h, 81]) = νexplicit[
              (Subscript[s, 81])] /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1 /. (Subscript[ν, 2]) -> (Subscript[ν, 2]) - 
              1 /. d -> d - 2

(Subscript[f, 81]) = 
      MakeFun[(Subscript[h, 81]), "(81)", 
        IFF[PP =!= 0 && (Subscript[ν, 1]) > 1 && (Subscript[ν, 2]) > 1 && 
            MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTJIRecursion[(Subscript[f, 81])]

MakeFun (8123)

(Subscript[h, 8123]) = Interchange[(Subscript[s, 81]), 
              2 ↔ 3] /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 
                1 /. (Subscript[ν, 3]) -> (Subscript[ν, 3]) - 1 /. d -> d - 2

(Subscript[f, 8123]) = 
      MakeFun[(Subscript[h, 8123]), "(8123)", 
        IFF[PP =!= 0 && (Subscript[ν, 1]) > 1 && (Subscript[ν, 3]) > 1 && 
            MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTJIRecursion[(Subscript[f, 8123])]

MakeFun (8113)

(Subscript[h, 8113]) = Interchange[(Subscript[s, 81]), 
              1 ↔ 3] /. (Subscript[ν, 2]) -> (Subscript[ν, 2]) - 
                1 /. (Subscript[ν, 3]) -> (Subscript[ν, 3]) - 1 /. 
        d -> d - 2

(Subscript[f, 8113]) = 
      MakeFun[(Subscript[h, 8113]), "(8113)", 
        IFF[PP =!= 0 && (Subscript[ν, 2]) > 1 && (Subscript[ν, 3]) > 1 && 
            MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTJIRecursion[(Subscript[f, 8113])]

Input (82)

general (Subscript[m, i]^2)and (Subscript[ν, 1]) > 0

(Subscript[e, 82])=-(p^2)(Subscript[m, 1]^2)((Subscript[Σ, 3]) + 
                  2)(Subscript[ν, 1])((Subscript[ν, 1]) + 1)\(1\^+\)\(1\^+\)(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d + 2)))+(1/2)(p^2)((Subscript[Σ, 3]) + 2)((Subscript[Σ, 3]) + 
                  4)(Subscript[ν, 1])\(1\^+\)(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d + 2)))==(-(1/2)((((p^2)))^2)(d + 2  (Subscript[ν, 1]) - 2  (Subscript[ν, 2]) - 2  (Subscript[ν, 3]))+(1/2)(p^2)((Subscript[m, 1]^2) (6  (Subscript[ν, 1]) + 6  (Subscript[ν, 2]) + 
                            6  (Subscript[ν, 3]) - 7  d) + (Subscript[m, 2]^2) (7  d - 
                            2  (Subscript[ν, 1]) - 4  (Subscript[ν, 2]) - 
                            10  (Subscript[ν, 3])) + (Subscript[m, 3]^2) (7  d - 
                            2  (Subscript[ν, 1]) - 10  (Subscript[ν, 2]) - 4  (Subscript[ν, 3])))+((((Subscript[m, 2]^2)))^2) (d - 
                      2  (Subscript[ν, 1]) - (Subscript[ν, 2]))+((((Subscript[m, 3]^2)))^2) (d - 
                      2  (Subscript[ν, 1]) - (Subscript[ν, 3]))+(Subscript[m, 1]^2) (Subscript[m, 2]^2) (d - (Subscript[ν, 1]) - 
                      2  (Subscript[ν, 2]))+(Subscript[m, 1]^2) (Subscript[m, 3]^2) (d - (Subscript[ν, 1]) - 
                      2  (Subscript[ν, 3]))-4  
                      (Subscript[m, 2]^2) (Subscript[m, 3]^2) (d - (Subscript[ν, 1]) - (Subscript[ν, 2]) - (Subscript[ν, 3]))-(Subscript[m, 1]^2)(2((((p^2)))^2)-(p^2)(2  (Subscript[m, 1]^2) + (Subscript[m, 2]^2) + (Subscript[m, 3]^2))+(Subscript[m, 1]^2) ((Subscript[m, 2]^2) + (Subscript[m, 3]^2))+3 ((((Subscript[m, 2]^2) - (Subscript[m, 3]^2)))^2))(Subscript[ν, 1])\(1\^+\)+(Subscript[m, 2]^2)(((((p^2)))^2)+((p^2)-(Subscript[m, 2]^2))(3  (Subscript[m, 1]^2)-5  (Subscript[m, 3]^2)+\(1\^-\)-\(3\^-\))-((((Subscript[m, 2]^2)))^2))(Subscript[ν, 2])\(2\^+\)+(Subscript[m, 3]^2)(((((p^2)))^2)+((p^2)-(Subscript[m, 3]^2))(3  (Subscript[m, 1]^2)-5  (Subscript[m, 2]^2)+\(1\^-\)-\(2\^-\))-((((Subscript[m, 3]^2)))^2))(Subscript[ν, 3])\(3\^+\)-(1/2)(p^2)((Subscript[Σ, 3]) + 2)(\(1\^-\)-\(2\^-\)-\(3\^-\))-(Subscript[m, 1]^2)((Subscript[m, 2]^2) - (Subscript[m, 3]^2))(Subscript[ν, 1])\(1\^+\)(\(2\^-\)-\(3\^-\)))(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)));

(Subscript[r, 82]) = OperatorApplyJ[(Subscript[e, 82])]

(Subscript[r, 82])[[1, 1]]

(Subscript[s, 82]) = ((Subscript[r, 82])[[1, 
            2]] == -(Subscript[r, 82])[[1, 1]] + 
            (Subscript[r, 82])[[2]]);

MakeFun (82)

(Subscript[h, 82]) = νexplicit[(Subscript[s, 82])] /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 2 /. 
        d -> d - 2

(Subscript[f, 82]) = 
      MakeFun[(Subscript[h, 82]), "(82)", 
        IFF[PP =!= 0 && (Subscript[ν, 1]) > 2 && m1 =!= 0 && 
            MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTJIRecursion[(Subscript[f, 82])]

MakeFun (8213)

(Subscript[h, 8213]) = Interchange[(Subscript[s, 82]), 
            1 ↔ 3] /. (Subscript[ν, 3]) -> (Subscript[ν, 3]) - 2 /. 
        d -> d - 2

(Subscript[f, 8213]) = 
      MakeFun[(Subscript[h, 8213]), "(8213)", 
        IFF[PP =!= 0 && m3 =!= 0 && (Subscript[ν, 3]) > 2 && 
            MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTJIRecursion[(Subscript[f, 8213])]

MakeFun (8212)

(Subscript[h, 8212]) = Interchange[(Subscript[s, 82]), 
            1 ↔ 2] /. (Subscript[ν, 2]) -> (Subscript[ν, 2]) - 2 /. 
        d -> d - 2

(Subscript[f, 8212]) = 
      MakeFun[(Subscript[h, 8212]), "(8212)", 
        IFF[PP =!= 0 && (Subscript[ν, 2]) > 2 && m2 =!= 0 && 
            MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTJIRecursion[(Subscript[f, 8212])]

Input (83)

(Subscript[m, 1]^2) == 0 and (Subscript[ν, 1]) > 0

(Subscript[e, 83])=(p^2)((Subscript[Σ, 3]) + 2)((Subscript[Σ, 3]) + 
                4)(Subscript[ν, 1])\(1\^+\)(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d + 2)))==(-((((p^2)))^2)(d + 2  (Subscript[ν, 1]) - 
                      2  (Subscript[ν, 2]) - 2  (Subscript[ν, 3]))+(p^2)((Subscript[m, 2]^2) (7  d - 2  (Subscript[ν, 1]) - 4  (Subscript[ν, 2]) - 
                            10  (Subscript[ν, 3])) + (Subscript[m, 3]^2) (7  d - 
                            2  (Subscript[ν, 1]) - 10  (Subscript[ν, 2]) - 4  (Subscript[ν, 3])))+2 ((((Subscript[m, 2]^2)))^2) (d - 
                      2  (Subscript[ν, 1]) - (Subscript[ν, 2]))+2 ((((Subscript[m, 3]^2)))^2) (d - 
                      2  (Subscript[ν, 1]) - (Subscript[ν, 3]))-8  
                      (Subscript[m, 2]^2) (Subscript[m, 3]^2) (d - (Subscript[ν, 1]) - (Subscript[ν, 2]) - (Subscript[ν, 3]))+2(Subscript[m, 2]^2)(((((p^2)))^2)-((p^2)-(Subscript[m, 2]^2))(5  (Subscript[m, 3]^2)-\(1\^-\)+\(3\^-\))-((((Subscript[m, 2]^2)))^2))(Subscript[ν, 2])\(2\^+\)+2(Subscript[m, 3]^2)(((((p^2)))^2)-((p^2)-(Subscript[m, 3]^2))(5  (Subscript[m, 2]^2)-\(1\^-\)+\(2\^-\))-((((Subscript[m, 3]^2)))^2))(Subscript[ν, 3])\(3\^+\)-(p^2)((Subscript[Σ, 3]) + 2)(\(1\^-\)-\(2\^-\)-\(3\^-\)))(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)));

(Subscript[s, 83]) = OperatorApplyJ[(Subscript[e, 83])]

MakeFun (83)

(Subscript[h, 83]) = νexplicit[(Subscript[s, 83])] /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1 /. 
        d -> d - 2

(Subscript[f, 83]) = 
      MakeFun[(Subscript[h, 83]), "(83)", 
        IFF[PP =!= 0 && (Subscript[ν, 1]) > 1 && 
            MatchQ[d, _Symbol + _Integer?Positive]], m1 :> 0]

CheckTJIRecursion[(Subscript[f, 83])]

Input (87)

(88)

(Subscript[t, 1]) = (Subscript[m, 1]^2) + (Subscript[m, 2]^2) + (Subscript[m, 3]^2);

(Subscript[t, 2]) = 
      3 ((Subscript[m, 1]^4) + (Subscript[m, 2]^4) + (Subscript[m, 3]^4)) + 
        2 ((Subscript[m, 1]^2) (Subscript[m, 2]^2) + (Subscript[m, 1]^2) (Subscript[m, 3]^2) + (Subscript[m, 2]^2) 
                (Subscript[m, 3]^2));

(Subscript[t, 3]) = (Subscript[m, 1]^2) ((Subscript[m, 1]^4) - (Subscript[m, 2]^4) - 
              (Subscript[m, 3]^4)) + (Subscript[m, 2]^2) ((Subscript[m, 2]^4) - (Subscript[m, 1]^4) - 
              (Subscript[m, 3]^4)) + (Subscript[m, 3]^2) ((Subscript[m, 3]^4) - (Subscript[m, 1]^4) - (Subscript[m, 2]^4)) + 
        10  (Subscript[m, 1]^2) (Subscript[m, 2]^2) (Subscript[m, 3]^2);

f[μ1_, μ2_,  μ3_]:=μ1((p^2)-μ1)(-2(d - 4)((((p^2)))^2)+(p^2)(4  (Subscript[t, 1]) (5  d - 18) - 
                    24  μ1 (2  d - 7))-2 (4  d - 13) ((Subscript[t, 1])^2)+2 (9  d - 31) (Subscript[t, 2])-24  μ2 μ3 (4  d - 13)-24 μ(1^2) (2  d - 7));

(Subscript[f, 1, 2, 3]) = f[(Subscript[m, 1]^2), (Subscript[m, 2]^2), (Subscript[m, 3]^2)];

(Subscript[f, 2, 1, 3]) = f[(Subscript[m, 2]^2), (Subscript[m, 1]^2), (Subscript[m, 3]^2)];

(Subscript[f, 3, 2, 1]) = f[(Subscript[m, 3]^2), (Subscript[m, 2]^2), (Subscript[m, 1]^2)];

(89)

g[μ1_, μ2_,  μ3_]:=((μ1 μ2)/(d - 4))(4(d - 4)((((p^2)))^2)-4(7  d - 24)(p^2)(3  μ3 - 2  (Subscript[t, 1]))-((Subscript[t, 1])^2) (23  d - 80)+(Subscript[t, 2]) (9  d - 32)-12  μ(3^2) (d - 4)+12  μ1 μ2 (7  d - 24));

(Subscript[g, 1, 2, 3]) = g[(Subscript[m, 1]^2), (Subscript[m, 2]^2), (Subscript[m, 3]^2)];

(Subscript[g, 1, 3, 2]) = g[(Subscript[m, 1]^2), (Subscript[m, 3]^2), (Subscript[m, 2]^2)];

(Subscript[g, 2, 3, 1]) = g[(Subscript[m, 2]^2), (Subscript[m, 3]^2), (Subscript[m, 1]^2)];

(Subscript[g, 1, 2, 3])

general  (Subscript[m, i]^2)

(Subscript[e, 87])=3(p^2)(d - 3)(d - 4)(3  d - 8)(3  d - 10)(Subscript[J, 111]^((d)))==((((d - 4))^2)((((p^2)))^3)-2((((p^2)))^2)(Subscript[t, 1])(d - 4)(6  d - 23)+(p^2)(5  ((Subscript[t, 1])^2) (15  (d^2) - 117  d + 
                            224) - (Subscript[t, 2]) (42  (d^2) - 331  d + 
                            640))-(1/4) (d - 
                      5) ((Subscript[t, 3]) (27  d - 
                            90) - (Subscript[t, 1]) (Subscript[t, 2]) (3  d - 2) - 
                      2  ((Subscript[t, 1])^3) (5  d - 26))+(Subscript[f, 1, 2, 3])\(1\^+\)+(Subscript[f, 2, 1, 3])\(2\^+\)+(Subscript[f, 3, 2, 1])\(3\^+\)+(Subscript[g, 1, 2, 3])\(1\^+\)\(2\^+\)\(3\^-\)+(Subscript[g, 1, 3, 2])\(1\^+\)\(2\^-\)\(3\^+\)+(Subscript[g, 2, 3, 1])\(1\^-\)\(2\^+\)\(3\^+\))(Subscript[J, 111]^((d - 2)));

(Subscript[s, 87]) = OperatorApplyJ[(Subscript[e, 87])]

MakeFun (87)

(Subscript[h, 87]) = (Subscript[s, 87])

(Subscript[f, 87]) = 
      MakeFun[(Subscript[h, 87]), "(87)", 
        IFF[PP =!= 0 && MatchQ[d, _Symbol + _Integer?Positive]]];

CheckTJIRecursion not applicable.

Input (90)

general  (Subscript[m, i]^2)

(Subscript[e, 90])=3(p^2)(d - 3)(d - 4)(3  d - 10)\(1\^+\)(Subscript[J, 111]^((d)))==(-(Subscript[m, 1]^2)(((((p^2)))^2)(7  d - 24)-2(p^2)((4  d - 15) (Subscript[t, 1]) - (d - 5) 
                              (Subscript[m, 1]^2))+(3/2) (d - 5) ((Subscript[t, 1])^2)+(5/2) (d - 3) (Subscript[t, 2])-2 (5  d - 17) ((((Subscript[m, 1]^2)))^2)-2 (13  d - 45) (Subscript[m, 2]^2) (Subscript[m, 3]^2))\(1\^+\)+
				2(Subscript[m, 2]^2)((p^2)-(Subscript[m, 2]^2))((d - 3)((p^2)+(Subscript[m, 2]^2)-5  (Subscript[m, 3]^2))+(7  d - 25) (Subscript[m, 1]^2))\(2\^+\)+
				2(Subscript[m, 3]^2)((p^2)-(Subscript[m, 3]^2))((d - 3)((p^2)-5  (Subscript[m, 2]^2)+(Subscript[m, 3]^2))+(7  d - 25) (Subscript[m, 1]^2))\(3\^+\)-
				(d - 3)(d - 4)((((p^2)))^2)+(p^2)((7  d - 30) (d - 3) 
                        (Subscript[t, 1]) - (7  d - 31) (3  d - 10) (Subscript[m, 1]^2))+(1/4) (d - 
                      5) ((17  d - 66) ((Subscript[t, 1])^2) - (3  d - 14) 
                        (Subscript[t, 2]) - 
                      4 (3  d - 10) (((((Subscript[m, 1]^2)))^2) + 
                            5  (Subscript[m, 2]^2) (Subscript[m, 3]^2)))+	(2/(d - 4))((((p^2)+(Subscript[m, 2]^2))(7  d - 24)+(d - 4) (Subscript[m, 1]^2)-(5  d - 18) (Subscript[m, 3]^2))(Subscript[m, 1]^2)(Subscript[m, 2]^2)\(1\^+\)\(2\^+\)\(3\^-\)+
						(((p^2)+(Subscript[m, 3]^2))(7  d - 24)+(d - 4) (Subscript[m, 1]^2)-(5  d - 18) (Subscript[m, 2]^2))(Subscript[m, 1]^2)(Subscript[m, 3]^2)\(1\^+\)\(2\^-\)\(3\^+\)-
						2(((p^2)+(Subscript[m, 2]^2)+(Subscript[m, 3]^2))(d - 3)+(d - 4) (Subscript[m, 1]^2))(Subscript[m, 2]^2)(Subscript[m, 3]^2)\(1\^-\)\(2\^+\)\(3\^+\)))(Subscript[J, 111]^((d - 2)));

(Subscript[s, 90]) = OperatorApplyJ[(Subscript[e, 90])]

MakeFun (90)

(Subscript[h, 90]) = (Subscript[s, 90]);

(Subscript[f, 90]) = 
      MakeFun[(Subscript[h, 90]), "(90)", 
        IFF[PP =!= 0 && MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTJIRecursion not applicable.

Input (91)

general (Subscript[m, i]^2)

(Subscript[e, 91.1])= (d - 2)(Subscript[ν, 1])\(1\^+\)(Subscript[K, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==(-(Subscript[u, 1, 2, 3])-\(1\^-\)+\(2\^-\)+\(3\^-\))(Subscript[K, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d - 2)));

(Subscript[s, 91.1]) = OperatorApplyK[(Subscript[e, 91.1])]

(Subscript[e, 91.2])=(d - 2)(Subscript[ν, 2])\(2\^+\)(Subscript[ν, 3])\(3\^+\)(Subscript[K, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==(-2(Subscript[m, 1]^2)(Subscript[ν, 1])\(1\^+\)+(d - 2 - 2  (Subscript[ν, 1])))(Subscript[K, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d - 2)));

(Subscript[s, 91.2]) = OperatorApplyK[(Subscript[e, 91.2])]

(Subscript[e, 91.3])=(d - 2)(d - (Subscript[ν, 1]) - (Subscript[ν, 2]) - (Subscript[ν, 3]))(Subscript[K, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==-((Subscript[Δ, 1, 2, 3])+(Subscript[u, 1, 2, 3])\(1\^-\)+(Subscript[u, 2, 1, 3])\(2\^-\)+(Subscript[u, 3, 1, 2])\(3\^-\))(Subscript[K, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d - 2)));

(Subscript[s, 91.3]) = OperatorApplyK[(Subscript[e, 91.3])]

(Subscript[e, 91.4])=(Subscript[Δ, 1, 2, 3])(Subscript[ν, 1])\(1\^+\)(Subscript[K, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==((Subscript[u, 1, 2, 3]) (d - (Subscript[ν, 1]) - 2  (Subscript[ν, 2]))+2  (Subscript[m, 2]^2) ((Subscript[ν, 1]) - (Subscript[ν, 2]))+(Subscript[u, 3, 1, 2])(Subscript[ν, 1])\(1\^+\)(\(2\^-\)-\(3\^-\))+2(Subscript[m, 2]^2)(Subscript[ν, 2])\(2\^+\)(\(1\^-\)-\(3\^-\)))(Subscript[K, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)));

(Subscript[s, 91.4]) = OperatorApplyK[(Subscript[e, 91.4])]

MakeFun (911)

(Subscript[h, 911]) = (Subscript[s, 91.1]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1

(Subscript[f, 911]) = 
      MakeFun[(Subscript[h, 911]), "(911)", 
        IFF[(Subscript[ν, 1]) > 1 && 
            MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTJIRecursion not applicable.

MakeFun (912)

(Subscript[h, 912]) = (Subscript[s, 91.2]) /. (Subscript[ν, 2]) -> (Subscript[ν, 2]) - 
              1 /. (Subscript[ν, 3]) -> (Subscript[ν, 3]) - 1

(Subscript[f, 912]) = 
      MakeFun[(Subscript[h, 912]), "(912)", 
        IFF[(Subscript[ν, 2]) > 1 && (Subscript[ν, 3]) > 1 && 
            MatchQ[d, _Symbol + _Integer?Positive]]]

CheckTJIRecursion not applicable.

MakeFun (913)

(Subscript[h, 913]) = (Subscript[s, 91.3])

(Subscript[f, 913]) = 
      MakeFun[(Subscript[h, 913]), "(913)", 
        IFF[(n2 === 1  ||  n3 === 1) && 
            MatchQ[d, _Symbol + _Integer?Positive]], n1 :> 1]

CheckTJIRecursion not applicable.

MakeFun (914)

(Subscript[h, 914]) = (Subscript[s, 91.4]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1

(Subscript[f, 914]) = 
      MakeFun[(Subscript[h, 914]), "(914)", 
        IFF[(Subscript[Δ, 1, 2, 3]) =!= 0 && ((Subscript[ν, 1])  >  1 ) && 
            MatchQ[d, _Symbol]]]

CheckTJIRecursion not applicable.

Input (92)

(Subscript[Δ, 1, 2, 3]) = Cayley[1,2,3][{m1,m2,m3}]

Cayley[1, 2, 3][{m1, m2, m3}]

\(TraditionalForm\`\((\ 
        m\_1\%2 == \((m\_2 + m\_3)\)\^2\ )\)\ ⟹\ \((\ Δ\_\(1, 2, 3\) == 0\ )\)\)

(Subscript[e, 92])=2(Subscript[m, 2])(Subscript[m, 3])((Subscript[m, 2]) + (Subscript[m, 3]))(d - 2  (Subscript[ν, 1]) - 
                2  (Subscript[ν, 2]) - 2  (Subscript[ν, 3]) - 1)\(1\^+\)(Subscript[K, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==((Subscript[m, 2])((d - (Subscript[ν, 1]) - (Subscript[ν, 2]) - 2  (Subscript[ν, 3]) - 1)(\(1\^+\)\(2\^-\)-1)+((Subscript[ν, 1]) - (Subscript[ν, 2]) + 1)\(1\^+\)\(3\^-\))+
			(Subscript[m, 3])((d - (Subscript[ν, 1]) - 2  (Subscript[ν, 2]) - (Subscript[ν, 3]) - 1)(\(1\^+\)\(3\^-\)-1)+((Subscript[ν, 1]) - (Subscript[ν, 3]) + 1)\(1\^+\)\(2\^-\)))(Subscript[K, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)));

(Subscript[s, 92]) = OperatorApplyK[(Subscript[e, 92])]

Input (93)

( \(TraditionalForm\`\(\(\((m\_1 == 0)\)\  && \ \((m\_2 == 
                m\_3)\)\)\(\ \)\(\ \)\()\)\)\ ⟹\ \((\ Δ\_\(1, 2, 3\) == 0\ )\)\)

(Subscript[e, 93])=2(Subscript[m, 2]^2)(d - 2  (Subscript[ν, 1]) - 2)(d - 
                2  (Subscript[ν, 1]) - (Subscript[ν, 2]) - (Subscript[ν, 3]) - 1)(d - 
                2  (Subscript[ν, 1]) - (Subscript[ν, 2]) - (Subscript[ν, 3]))\(1\^+\)(Subscript[K, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==(d - 2  (Subscript[ν, 1]) - 2  (Subscript[ν, 3]))(d - 2  (Subscript[ν, 1]) - 
                2  (Subscript[ν, 2]))(d - (Subscript[ν, 1]) - (Subscript[ν, 2]) - (Subscript[ν, 3]))(Subscript[K, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)));

(Subscript[s, 93]) = OperatorApplyK[(Subscript[e, 93])]

MakeFun (93)

(Subscript[h, 93]) = (Subscript[s, 93]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1

(Subscript[f, 93]) = 
      MakeFun[(Subscript[h, 93]), "(93)", 
        IFF[(Subscript[ν, 1]) > 1 && (Subscript[m, 2]) =!= 0], {m1 :> 0, m3 :> m2}]

CheckTJIRecursion not applicable.

Input (94)

general (Subscript[m, i]^2)

(Subscript[e, 94])=(Subscript[Δ, 1, 2, 6])(Subscript[ν, 1])\(1\^+\)(Subscript[B, (Subscript[ν, 1])(Subscript[ν, 2])]^((d)))==((Subscript[u, 6, 1, 2])((Subscript[ν, 1])\(1\^+\)\(2\^-\)-d+(Subscript[ν, 1])+2  (Subscript[ν, 2]))+2(Subscript[m, 2]^2)((Subscript[ν, 2])\(2\^+\)\(1\^-\)-d+2  (Subscript[ν, 1])+(Subscript[ν, 2])))(Subscript[B, (Subscript[ν, 1])(Subscript[ν, 2])]^((d)));

(Subscript[s, 94]) = OperatorApplyB[(Subscript[e, 94])]

MakeFun (94)

(Subscript[h, 94]) = (Subscript[s, 94]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1

(Subscript[f, 94]) = 
      MakeFun[(Subscript[h, 94]), "(94)", 
        IFF[(Subscript[Δ, 1, 2, 6]) =!= 0 && (Subscript[ν, 1]) > 1]]

MakeFun (9401)

(Subscript[n, 9401]) = (Subscript[s, 94]) /. (Subscript[Δ, 1, 2, 6]) -> 0 /. 
        (Subscript[u, 6, 1, 2]) -> 0

(Subscript[h, 9401]) = -(Subscript[n, 9401])[[2, 2]] == 
        (Subscript[n, 9401])[[2, 1]]

(Subscript[f, 9401]) = 
      MakeFun[(Subscript[h, 9401]), "(9401)", 
        IFF[(Subscript[Δ, 1, 2, 6]) === 0 && (Subscript[u, 6, 1, 2]) === 0 && 
            m2 =!= 0]]

MakeFun (9402)

(Subscript[n, 9402]) = 
      Interchange[(Subscript[s, 94]) /. (Subscript[Δ, 1, 2, 6]) -> 0 /. 
          (Subscript[u, 6, 1, 2]) -> 0, 1 ↔ 2]

(Subscript[h, 9402]) = -(Subscript[n, 9402])[[2, 1]] == 
        (Subscript[n, 9402])[[2, 2]]

(Subscript[f, 9402]) = 
      MakeFun[(Subscript[h, 9402]), "(9402)", 
        IFF[(Subscript[Δ, 1, 2, 6]) === 0 && (Subscript[u, 6, 1, 2]) === 0 && 
            m1 =!= 0]]

Input (95)

general (Subscript[m, i]^2)

(Subscript[e, 95])=2(p^2)(Subscript[ν, 1])\(1\^+\)(Subscript[B, (Subscript[ν, 1])(Subscript[ν, 2])]^((d)))==((Subscript[u, 1, 2, 6])+\(1\^-\)-\(2\^-\))(Subscript[B, (Subscript[ν, 1])(Subscript[ν, 2])]^((d - 2)));

(Subscript[s, 95]) = OperatorApplyB[(Subscript[e, 95])]

MakeFun (95)

(Subscript[h, 95]) = (Subscript[s, 95]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1

(Subscript[f, 95]) = 
      MakeFun[(Subscript[h, 95]), "(95)", 
        IFF[(Subscript[ν, 1]) > 1  &&  
            MatchQ[d, _Symbol + _Integer?Positive]]]

Input (96)

general (Subscript[m, i]^2)

(Subscript[e, 96])=2(p^2)(d - (Subscript[ν, 1]) - (Subscript[ν, 2]) - 1)(Subscript[B, (Subscript[ν, 1])(Subscript[ν, 2])]^((d)))==((Subscript[Δ, 1, 2, 6])+(Subscript[u, 1, 2, 6])\(1\^-\)+(Subscript[u, 2, 1, 6])\(2\^-\))(Subscript[B, (Subscript[ν, 1])(Subscript[ν, 2])]^((d - 2)));

(Subscript[s, 96]) = OperatorApplyB[(Subscript[e, 96])]

MakeFun (96)

(Subscript[h, 96]) = (Subscript[s, 96])

(Subscript[f, 96]) = 
      MakeFun[(Subscript[h, 96]), "(96)", 
        IFF[ MatchQ[d, _Symbol + _Integer ?Positive]], n1 :> 1]

Input (97)

general (Subscript[m, i]^2)

(Subscript[e, 97])=2(Subscript[m, 1]^2)(Subscript[ν, 1])\(1\^+\)(Subscript[A, (Subscript[ν, 1])]^((d)))==(d - 2  (Subscript[ν, 1]))(Subscript[A, (Subscript[ν, 1])]^((d)));

(Subscript[s, 97]) = OperatorApplyA[(Subscript[e, 97])]

MakeFun (97)

(Subscript[h, 97]) = (Subscript[s, 97]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1

(Subscript[f, 97]) = 
      MakeFun[(Subscript[h, 97]), "(97)", 
        IFF[(Subscript[ν, 1]) > 1 &&  MatchQ[d, _Symbol ]]]

Input (98)

general (Subscript[m, i]^2)

(Subscript[e, 98])=(d - 2  (Subscript[ν, 1]))(Subscript[A, (Subscript[ν, 1])]^((d)))==-2(Subscript[m, 1]^2)(Subscript[A, (Subscript[ν, 1])]^((d - 2)));

(Subscript[s, 98]) = OperatorApplyA[(Subscript[e, 98])]

MakeFun (98)

(Subscript[h, 98]) = (Subscript[s, 98])

(Subscript[f, 98]) = 
      MakeFun[(Subscript[h, 98]), "(98)", 
        IFF[MatchQ[d, _Symbol  + _Integer?Positive]]]

Input (99)

(Subscript[m, 1]) = M,  (p^2) = (M^2), 
    (Subscript[m, 2]) = (Subscript[m, 3]) = 0

Herleitung in three.nb -(Subscript[ν, 1]) (d - 2 (Subscript[ν, 2]) - 2 (Subscript[ν, 3])) (d - (Subscript[ν, 2]) - (Subscript[ν, 3]) - 1) (Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==(Subscript[ν, 2]) (-d + 2 (Subscript[ν, 2]) + 2) (-2 d + (Subscript[ν, 1]) + 2 (Subscript[ν, 2]) + 2 (Subscript[ν, 3]) + 2) (Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))

(Subscript[e, 99])=(Subscript[ν, 1]) (d - 2 (Subscript[ν, 2]) - 2 (Subscript[ν, 3])) (d - (Subscript[ν, 2]) - (Subscript[ν, 3]) - 1) \(1\^+\)(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==-(Subscript[ν, 2]) (-d + 2 (Subscript[ν, 2]) + 2) (-2 d + (Subscript[ν, 1]) + 2 (Subscript[ν, 2]) + 2 (Subscript[ν, 3]) + 2)\(2\^+\)(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))

(Subscript[s, 99]) = OperatorApplyJ[(Subscript[e, 99])]

MakeFun (99)

(Subscript[h, 99]) = (Subscript[s, 99]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1

(Subscript[f, 99]) = 
      MakeFun[(Subscript[h, 99]), "(99)", 
        IFF[m1^2 === PP && (Subscript[ν, 1]) > 1], {m2 :> 0, m3 :> 0}]

MakeFun (9912)

(Subscript[h, 9912]) = 
      Interchange[(Subscript[s, 99]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1, 
        1 ↔ 2]

(Subscript[f, 9912]) = 
      MakeFun[(Subscript[h, 9912]), "(9912)", 
        IFF[m2^2 === PP && (Subscript[ν, 2]) > 1], {m1 :> 0, m3 :> 0}]

MakeFun (9913)

(Subscript[h, 9913]) = 
      Interchange[(Subscript[s, 99]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1, 
        1 ↔ 3]

(Subscript[f, 9913]) = 
      MakeFun[(Subscript[h, 9913]), "(9913)", 
        IFF[m3^2 === PP && (Subscript[ν, 3]) > 1], {m1 :> 0, m2 :> 0}]

Input (100)

Herleitung in three.nb

2 (Subscript[m, 2]^2) (1 - (Subscript[ν, 2])) (Subscript[ν, 2]) (Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==(d - 2 (Subscript[ν, 2])) (1 - (Subscript[ν, 2])) (Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))-(Subscript[ν, 1]) (-d + 2 (Subscript[ν, 1]) + 2) (Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))

(Subscript[m, 1]) = 0

(Subscript[e, 100])=2 (Subscript[m, 2]^2) (1 - (Subscript[ν, 2])) (Subscript[ν, 2]) \(2\^+\)(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==(d - 2 (Subscript[ν, 2])) (1 - (Subscript[ν, 2])) (Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))-(Subscript[ν, 1]) (-d + 2 (Subscript[ν, 1]) + 2) \(1\^+\)\(2\^-\)(Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))

(Subscript[s, 100]) = OperatorApplyJ[(Subscript[e, 100])]

MakeFun (100)

(Subscript[h, 100]) = (Subscript[s, 100])

(Subscript[f, 100]) = 
      MakeFun[(Subscript[h, 100]) /. (Subscript[ν, 2]) -> (Subscript[ν, 2]) - 1, "(100)", 
        IFF[(Subscript[ν, 2]) > 2 && m2 =!= 0], m1 :> 0]

CheckTJIRecursion[(Subscript[f, 100])]

MakeFun (1002)

(Subscript[h, 1002]) = Interchange[(Subscript[s, 100]), 1 ↔ 3]

(Subscript[f, 1002]) = 
      MakeFun[(Subscript[h, 1002]) /. (Subscript[ν, 2]) -> (Subscript[ν, 2]) - 1, "(1002)", 
        IFF[(Subscript[ν, 2]) > 2 && m2 =!= 0], m3 :> 0]

CheckTJIRecursion[(Subscript[f, 1002])]

MakeFun (1003)

(Subscript[h, 1003]) = 
      Interchange[(Subscript[s, 100]), 1 ↔ 3, 
        1 ↔ 2]

(Subscript[f, 1003]) = 
      MakeFun[(Subscript[h, 1003]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1, "(1003)", 
        IFF[(Subscript[ν, 1]) > 2 && m1 =!= 0], m3 :> 0]

CheckTJIRecursion[(Subscript[f, 1003])]

MakeFun (1004)

(Subscript[h, 1004]) = Interchange[(Subscript[s, 100]), 1 ↔ 2]

(Subscript[f, 1004]) = 
      MakeFun[(Subscript[h, 1004]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1, "(1004)", 
        IFF[(Subscript[ν, 1]) > 2 && m1 =!= 0], m2 :> 0]

CheckTJIRecursion[(Subscript[f, 1004])]

MakeFun (1005)

(Subscript[h, 1005]) = Interchange[(Subscript[s, 100]), 2 ↔ 3]

(Subscript[f, 1005]) = 
      MakeFun[(Subscript[h, 1005]) /. (Subscript[ν, 3]) -> (Subscript[ν, 3]) - 1, "(1005)", 
        IFF[(Subscript[ν, 3]) > 2 && m3 =!= 0], m1 :> 0]

CheckTJIRecursion[(Subscript[f, 1005])]

MakeFun (1006)

(Subscript[h, 1006]) = 
      Interchange[(Subscript[s, 100]), 2 ↔ 3, 
        1 ↔ 2]

(Subscript[f, 1006]) = 
      MakeFun[(Subscript[h, 1006]) /. (Subscript[ν, 3]) -> (Subscript[ν, 3]) - 1, "(1006)", 
        IFF[(Subscript[ν, 3]) > 2 && m3 =!= 0], m2 :> 0]

CheckTJIRecursion[(Subscript[f, 1006])]

Input (150)

(Subscript[ν, 4]) = -1

((((Subscript[q, 2]) - p ))^2) - (Subscript[m, 4]^2) =  
      (Subscript[q, 2]^2) - (Subscript[m, 2]^2) - 2  p (Subscript[q, 2]) + (p^2) + (Subscript[m, 2]^2) - (Subscript[m, 4]^2)

(Subscript[s, 150]) = 
      TFI[d, PP, {{n1, m1}, {n2, m2}, {n3, m3}, {-1, m4}, {n5, m5}}] == 
        TFI[d, PP, {{n1, m1}, {n2 - 1, m2}, {n3, m3}, {0, 0}, {n5, m5}}] - 
          2  TFI[d, 
              PP, {0, 0, 0, 1, 
                0}, {{n1, m1}, {n2, m2}, {n3, m3}, {0, 0}, {n5, 
                  m5}}] + (PP + m2^2 - m4^2) 
            TFI[d, PP, {{n1, m1}, {n2, m2}, {n3, m3}, {0, 0}, {n5, 
                  m5}}]

MakeFun (150)

(Subscript[h, 150]) = (Subscript[s, 150])

(Subscript[f, 150]) = MakeFun[(Subscript[h, 150]), "(150)"]

Input (151)

(Subscript[ν, 4]) = -1

((((Subscript[q, 2]) - p ))^2) - (Subscript[m, 4]^2) =  
      (Subscript[q, 2]^2) - (Subscript[m, 2]^2) - 2  p (Subscript[q, 2]) + (p^2) + (Subscript[m, 2]^2) - (Subscript[m, 4]^2)

(Subscript[s, 151]) = 
      TFI[d, PP, 
          DP, {a, b}, {{n1, m1}, {n2, m2}, {n3, m3}, {-1, m4}, {n5, 
              m5}}] == TFI[d, PP, 
            DP, {a, b}, {{n1, m1}, {n2 - 1, m2}, {n3, m3}, {0, 0}, {n5, 
                m5}}] - 
          2  TFI[d, PP, 
              DP, {a, b}, {0, 0, 0, 1, 
                0}, {{n1, m1}, {n2, m2}, {n3, m3}, {0, 0}, {n5, 
                  m5}}] + (PP + m2^2 - m4^2) 
            TFI[d, PP, 
              DP, {a, b}, {{n1, m1}, {n2, m2}, {n3, m3}, {0, 0}, {n5, 
                  m5}}]

MakeFun (151)

(Subscript[h, 151]) = (Subscript[s, 151])

(Subscript[f, 151]) = 
      ReplacePart[
        MakeFun[(Subscript[h, 151]), "(151)"], {a_, b_}, {1, 1, 4}]

Input (152)

(Subscript[ν, 4]) = -1

((((Subscript[q, 2]) - p ))^2) - (Subscript[m, 4]^2) =  
      (Subscript[q, 2]^2) - (Subscript[m, 2]^2) - 2  p (Subscript[q, 2]) + (p^2) + (Subscript[m, 2]^2) - (Subscript[m, 4]^2)

(Subscript[s, 152]) = 
      TVI[d, PP, {{n1, m1}, {n2, m2}, {n3, m3}, {-1, m4}}] == 
        TVI[d, PP, {{n1, m1}, {n2 - 1, m2}, {n3, m3}, {0, 0}}] - 
          2  TFI[d, 
              PP, {0, 0, 0, 1, 
                0}, {{0, 0}, {n2, m2}, {n3, m3}, {0, 0}, {n1, m1}}] + (PP + 
                m2^2 - m4^2) 
            TVI[d, PP, {{n1, m1}, {n2, m2}, {n3, m3}, {0, 0}}]

MakeFun (152)

(Subscript[h, 152]) = (Subscript[s, 152])

(Subscript[f, 152]) = MakeFun[(Subscript[h, 152]), "(152)"]

Input (153)

(Subscript[ν, 2]) = -1

\(s\_153 = 
      TVI[d, PP, {{n1, m1}, {\(-1\), m2}, {n3, m3}, {n4, m4}}] == 
		TFI[
            d, \ PP, {0, 1, 0, 0, 
              0}, \ {{0, 0}, {0, 0}, {n3, m3}, {n4, m4}, {n1, m1}}] + 
          m2^2\ TVI[d, \ 
              PP, \ {{n1, m1}, {0, 0}, {n3, m3}, {n4, m4}}]\)

MakeFun (153)

(Subscript[h, 153]) = (Subscript[s, 153])

(Subscript[f, 153]) = MakeFun[(Subscript[h, 153]), "(153)"]

Input (200)

Derivation in TJRspecial.nb

(Subscript[m, 3]) = (Subscript[m, 2]) = (Subscript[m, 1]),  (p^2) = (Subscript[m, 1]^2)

(Subscript[s, 200])=16 (Subscript[m, 1]^2) (Subscript[ν, 1]) (d - (Subscript[ν, 1]) - (Subscript[ν, 2]) - (Subscript[ν, 3])) (Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))==-(1 + 3 d - 3 (Subscript[ν, 1]) - 4 (Subscript[ν, 2])) (Subscript[ν, 3]) (Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))-(Subscript[ν, 2]) (1 + 3 d - 3 (Subscript[ν, 1]) - 4 (Subscript[ν, 3])) (Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))+(-1 + 2 d - (Subscript[ν, 1]) - 2 (Subscript[ν, 2])) (Subscript[ν, 3]) (Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))-(-6 (d^2) + 22 d (Subscript[ν, 1]) - 
                  16 (Subscript[ν, 1]^2) - (Subscript[ν, 2]) + 6 d (Subscript[ν, 2]) - 
                  13 (Subscript[ν, 1]) (Subscript[ν, 2]) - (Subscript[ν, 3]) + 6 d (Subscript[ν, 3]) - 
                  13 (Subscript[ν, 1]) (Subscript[ν, 3]) - 4 (Subscript[ν, 2]) (Subscript[ν, 3])) (Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)))+(Subscript[ν, 2]) (-1 + 2 d - (Subscript[ν, 1]) - 2 (Subscript[ν, 3])) (Subscript[J, (Subscript[ν, 1])(Subscript[ν, 2])(Subscript[ν, 3])]^((d)));

\(TextForm\`Due\ to\ sorting\ \ ν\_1 ⩾ ν\_2 ⩾ ν\_3\ \ by\ TJI\ only\ this\ rule\ is\ \(\(necessary\)\(.\)\)\)

MakeFun (200)

(Subscript[h, 200]) = (Subscript[s, 200]) /. (Subscript[ν, 1]) -> (Subscript[ν, 1]) - 1

(Subscript[f, 200])=MakeFun[(Subscript[h, 200]),"(200)",IFF[(Subscript[ν, 1]) > 1 && m1 =!= 0 && PP === m1^2],{m2 :> m1, m3 :> m1}]

MakeFun (201)

Subscript[f, 201] = 
  HoldForm[TJR[d_, PP_, 
     {{(n1_)?PQ, 0}, 
      {(n2_)?PQ, 0}, 
      {(n3_)?PQ, m3_}}] := 
    (TComment["(201)", 
        TJIC[d, 
         {{n1, 0}, {n2, 0}, 
          {n3, m3}}]]; 
       -(((d - n1 - n2)*
          (-2 - d + 
         2*n1 + 2*n2)*
          (3*d - 2*n1 - 
         2*n2 - 2*n3)*
          (-1 - d + n1 + 
          n2 + n3)*
          TJI[d, PP, 
          {{-1 + n1, 0}, 
          {n2, 0}, {n3, m3}}]
)/(2*m3^2*(d - 2*n1)*
          (-1 + n1)*
          (2*d - 2*n1 - 
          2*n2 - n3)*
          (1 + 2*d - 2*n1 - 
          2*n2 - n3)))) /; 
     n1 > 1 && m3 =!= 0 && 
      PP === m3^2]

MakeFun (202)

Subscript[f, 202] = 
  HoldForm[TJR[d_, PP_, 
     {{(n1_)?PQ, 0}, 
      {(n3_)?PQ, m3_}, 
      {(n2_)?PQ, 0}}] := 
    (TComment["(202)", 
        TJIC[d, 
         {{n1, 0}, {n2, 0}, 
          {n3, m3}}]]; 
       -(((d - n1 - n2)*
          (-2 - d + 2*n1 + 
          2*n2)*
          (3*d - 2*n1 - 
          2*n2 - 2*n3)*
          (-1 - d + n1 + 
          n2 + n3)*
          TJI[d, m3^2, 
          {{-1 + n1, 0}, 
          {n2, 0}, {n3, m3}}
])/(2*m3^2*(d - 2*n1)*
          (-1 + n1)*
          (2*d - 2*n1 - 
          2*n2 - n3)*
          (1 + 2*d - 2*n1 - 
          2*n2 - n3)))) /; 
     n1 > 1 && m3 =!= 0 && 
      PP === m3^2]

MakeFun (203)

\(Subscript[f, \ 203]\  = 
      HoldForm[
		\ TJR[d_, \ 
            0, \ {{\((n1_)\)?PQ, \ 0}, \ {\((n2_)\)?PQ, \ 
                m2_}, \ 
\ \ \ \ {\((n3_)\)?PQ, \ 
                m2_}}]\  := \((
				\ TComment["(203)", \ 
\ \ \ \ \ \ \ \ TJIC[d, 
                0, \ {{n1, \ 0}, \ {n2, \ m2}, \ {n3, \ 
                    m2}}]]; \ 
				\((\((d\  - \ 
                      2*\((\(-1\)\  + \ n1\  + \ 
                            n2)\))\)*
\ \ \ \ \ \((1\  + \ d\  - \ n1\  - \ 
                      n2\  - \ n3)\)*
\ \ \ \ \ \((d\  - \ 
                      2*\((\(-1\)\  + \ n1\  + \ n3)\))\)*
\ \ \ \ \ TJR[
                    d, \ 0, \ {{\(-1\)\  + \ n1, \ 0}, \ {n2, \ 
                        m2}, \ 
\ \ \ \ \ \ \ {n3, \ m2}}])\)/
\ \ \ \((2*
                  m2^2*\((d\  - \ 2*n1)\)*
\ \ \ \ \ \((1\  + \ d\  - \ 
                      2*n1\  - \ n2\  - \ n3)\)*
\ \ \ \ \ \((2\  + \ 
                      d\  - \ 2*n1\  - \ n2\  - \ 
                      n3)\))\)
			
			)\)]\)

MakeFun (204)

\(Subscript[f, \ 204]\  = 
      HoldForm[
		\ TJR[d_, \ 
            0, \ {\ {\((n2_)\)?PQ, \ m2_}, \ {\((n1_)\)?PQ, \ 
                0}, 
\ \ \ \ {\((n3_)\)?PQ, \ 
                m2_}}]\  := \((
	\ TComment["(204)", \ 
\ \ \ \ \ \ \ \ TJIC[d, 0, \ {\ {n2, \ m2}, \ {n1, \ 0}, {n3, \ 
                    m2}}]]; \ 
	\((\((d\  - \ 
                      2*\((\(-1\)\  + \ n1\  + \ 
                            n2)\))\)*
\ \ \ \ \ \((1\  + \ d\  - \ n1\  - \ 
                      n2\  - \ n3)\)*
\ \ \ \ \ \((d\  - \ 
                      2*\((\(-1\)\  + \ n1\  + \ n3)\))\)*
\ \ \ \ \ TJR[
                    d, \ 0, \ {{\(-1\)\  + \ n1, \ 0}, \ {n2, \ 
                        m2}, \ 
\ \ \ \ \ \ \ {n3, \ m2}}])\)/
\ \ \ \((2*
                  m2^2*\((d\  - \ 2*n1)\)*
\ \ \ \ \ \((1\  + \ d\  - \ 
                      2*n1\  - \ n2\  - \ n3)\)*
\ \ \ \ \ \((2\  + \ 
                      d\  - \ 2*n1\  - \ n2\  - \ 
                      n3)\))\)
			)\)]\)

MakeFun (205)

\(Subscript[f, \ 205]\  = 
      HoldForm[
		\ TJR[d_, \ 
            0, \ {\ {\((n2_)\)?PQ, \ m2_}, 
\ \ \ \ {\((n3_)\)?PQ, \ 
                m2_}, \ {\((n1_)\)?PQ, \ 
                0}}]\  := \((
	\ TComment["(205)", \ 
\ \ \ \ \ \ \ \ TJIC[d, 0, \ {\ {n2, \ m2}, {n3, \ m2}, \ {n1, \ 
                    0}}]]; \ 
\((\((d\  - \ 
                      2*\((\(-1\)\  + \ n1\  + \ 
                            n2)\))\)*
\ \ \ \ \ \((1\  + \ d\  - \ n1\  - \ 
                      n2\  - \ n3)\)*
\ \ \ \ \ \((d\  - \ 
                      2*\((\(-1\)\  + \ n1\  + \ n3)\))\)*
\ \ \ \ \ TJR[
                    d, \ 0, \ {{\(-1\)\  + \ n1, \ 0}, \ {n2, \ 
                        m2}, \ 
\ \ \ \ \ \ \ {n3, \ m2}}])\)/
\ \ \ \((2*
                  m2^2*\((d\  - \ 2*n1)\)*
\ \ \ \ \ \((1\  + \ d\  - \ 
                      2*n1\  - \ n2\  - \ n3)\)*
\ \ \ \ \ \((2\  + \ 
                      d\  - \ 2*n1\  - \ n2\  - \ 
                      n3)\))\)
			)\)]\)

Input (1xxxx)

If[MatchQ[$RankLimit, {_Integer?NonNegative, 
     _Integer?NonNegative}], 
   $TarasovTdeltaplimit = $RankLimit[[1]]; 
    ($TarasovTplimit = $RankLimit[[2]]; ), 
   $TarasovTdeltaplimit = 2; $TarasovTplimit = 2];

tarti=Timing[Do[If[0 < a + b + r + es,WriteString["stdout", {a, b, r, es}];Set@@{(Subscript[s, (10^4) + 1000 a + 100 b + 10 r + 
                              es]),TFI[d,(p^2),DP,{a, b},{0, 0, r, es, 0},{{n1,(Subscript[m, 1])},{n2,(Subscript[m, 2])},{n3,(Subscript[m, 3])},{n4,(Subscript[m, 4])},{n5,(Subscript[m, 5])}}]==ApplyTarasovT[TarasovT[a,b,r,es,(p^2),DP],STLI[d,(p^2),{{n1,(Subscript[m, 1])},{n2,(Subscript[m, 2])},{n3,(Subscript[m, 3])},{n4,(Subscript[m, 4])},{n5,(Subscript[m, 5])}}]]/.STLI->TFI}],{a, 0, $TarasovTdeltaplimit},{b, 0, $TarasovTdeltaplimit - a},{r, 0, $TarasovTplimit},{es, 0, $TarasovTplimit - r}]]

Share[]

\(matarti = 
      Timing[Do[
          If[0 < ia + ib + ir + es, 
            WriteString["stdout", {ia, ib, ir, es}]; 
            Set[f\_\(10\^4 + 1000\ ia + 100\ ib + 10\ ir + es\), 
              MakeFun[s\_\(10\^4 + 1000\ ia + 100\ ib + 10\ ir + es\), "(1" <> ToString[ia] <> ToString[ib] <> ToString[ir] <> 
                    ToString[es] <> ")"] /. PQ :> PNQ]], {ia, 
            0, \ $TarasovTdeltaplimit}, {ib, 
            0, $TarasovTdeltaplimit - ia}, {ir, 0, \ \ $TarasovTplimit}, {es, 
            0, $TarasovTplimit - ir}]]\)

Share[]

symmetry

Uncomment this ...

TVR[d_, pp_, {a_, b_, c_, z_}] := 
  TVR[d, pp, {c, b, a, z}] /;  !OrderedQ[{a, c}]

special TVR reductions

TVR[2 + d_Symbol, pp_, {{1, m1_}, {1, m1_}, {2, 0}, {1, m1_}}] := 
   ((-6*m1^2 + 2*d*m1^2 + 3*pp)*TAI[d, 0, {{1, m1}}]^2)/
      (12*(-3 + d)*(-1 + d)*m1^2*pp) + 
     ((4*m1^2 - pp)*TAI[d, 0, {{1, m1}}]*TBI[d, pp, {{1, m1}, {1, m1}}])/
      (4*(-3 + d)*(-1 + d)*m1^2) - 
     ((6*m1^2 - 2*d*m1^2 - 7*pp + 2*d*pp)*
        TJI[d, pp, {{1, m1}, {1, m1}, {1, 0}}])/(6*(-2 + d)*(-1 + d)*pp) + 
     ((-4*m1^2 + pp)*(-6*m1^2 + 2*d*m1^2 + 3*pp)*
        TJI[d, pp, {{2, m1}, {1, m1}, {1, 0}}])/
      (6*(-3 + d)*(-2 + d)*(-1 + d)*pp) /; (m1=!=0)&&(pp=!=0);

TVR[4 + d_Symbol, pp_, {{1, m1_}, {1, m1_}, {2, 0}, {1, m1_}}] := 
   -((-96*d*m1^6 + 352*d^2*m1^6 - 384*d^3*m1^6 + 128*d^4*m1^6 - 
           192*m1^4*pp + 1008*d*m1^4*pp - 1632*d^2*m1^4*pp + 
           912*d^3*m1^4*pp - 144*d^4*m1^4*pp + 288*m1^2*pp^2 - 
           1536*d*m1^2*pp^2 + 2316*d^2*m1^2*pp^2 - 1086*d^3*m1^2*pp^2 + 
           66*d^4*m1^2*pp^2 + 72*d*pp^3 - 166*d^2*pp^3 + 81*d^3*pp^3 + 
           4*d^4*pp^3)*TAI[d, 0, {{1, m1}}]^2)/
      (36*(-1 + d)^2*d^2*(1 + d)*(-4 + 3*d)*(-2 + 3*d)*pp^2) + 
     ((4*m1^2 - pp)^2*TAI[d, 0, {{1, m1}}]*TBI[d, pp, {{1, m1}, {1, m1}}])/
      (4*(-1 + d)^2*d*(1 + d)) - 
     ((288*m1^8 - 1152*d*m1^8 + 1504*d^2*m1^8 - 768*d^3*m1^8 + 
          128*d^4*m1^8 - 816*m1^6*pp + 2480*d*m1^6*pp - 
          2224*d^2*m1^6*pp + 736*d^3*m1^6*pp - 80*d^4*m1^6*pp + 
          864*m1^4*pp^2 - 2196*d*m1^4*pp^2 + 1446*d^2*m1^4*pp^2 - 
          252*d^3*m1^4*pp^2 - 6*d^4*m1^4*pp^2 - 432*m1^2*pp^3 + 
          1086*d*m1^2*pp^3 - 895*d^2*m1^2*pp^3 + 315*d^3*m1^2*pp^3 - 
          44*d^4*m1^2*pp^3 + 4*d*pp^4 - 2*d^2*pp^4 - 4*d^3*pp^4 + 
          2*d^4*pp^4)*TJI[d, pp, {{1, m1}, {1, m1}, {1, 0}}])/
      (18*(-2 + d)*(-1 + d)^2*d*(1 + d)*(-4 + 3*d)*(-2 + 3*d)*pp^2) + 
     (m1^2*(4*m1^2 - pp)*(-96*m1^6 + 352*d*m1^6 - 384*d^2*m1^6 + 
          128*d^3*m1^6 + 240*m1^4*pp - 624*d*m1^4*pp + 384*d^2*m1^4*pp - 
          48*d^3*m1^4*pp - 192*m1^2*pp^2 + 420*d*m1^2*pp^2 - 
          162*d^2*m1^2*pp^2 - 18*d^3*m1^2*pp^2 + 72*pp^3 - 154*d*pp^3 + 
          81*d^2*pp^3 - 8*d^3*pp^3)*TJI[d, pp, {{2, m1}, {1, m1}, {1, 0}}])/
      (18*(-2 + d)*(-1 + d)^2*d*(1 + d)*(-4 + 3*d)*(-2 + 3*d)*pp^2) /; (m1=!=0)&&(pp=!=0);

TVR[4 + d_Symbol, pp_, {{1, m1_}, {1, m1_}, {3, 0}, {1, m1_}}] := 
   ((-72*d^2*m1^6 + 144*d^3*m1^6 - 88*d^4*m1^6 + 16*d^5*m1^6 - 
          240*d*m1^4*pp + 572*d^2*m1^4*pp - 398*d^3*m1^4*pp + 
          84*d^4*m1^4*pp - 2*d^5*m1^4*pp - 96*m1^2*pp^2 + 
          480*d*m1^2*pp^2 - 744*d^2*m1^2*pp^2 + 390*d^3*m1^2*pp^2 - 
          50*d^4*m1^2*pp^2 + 4*d^5*m1^2*pp^2 - 24*d*pp^3 + 54*d^2*pp^3 - 
          27*d^3*pp^3)*TAI[d, 0, {{1, m1}}]^2)/
      (24*(-3 + d)*(-1 + d)^2*d*(1 + d)*(-4 + 3*d)*(-2 + 3*d)*m1^2*pp^2) + 
     ((4*m1^2 - pp)^2*TAI[d, 0, {{1, m1}}]*TBI[d, pp, {{1, m1}, {1, m1}}])/
      (8*(-3 + d)*(-1 + d)^2*(1 + d)*m1^2) + 
     ((-72*d*m1^6 + 144*d^2*m1^6 - 88*d^3*m1^6 + 16*d^4*m1^6 - 
          96*m1^4*pp + 260*d*m1^4*pp - 166*d^2*m1^4*pp + 
          12*d^3*m1^4*pp + 6*d^4*m1^4*pp + 160*m1^2*pp^2 - 
          300*d*m1^2*pp^2 + 50*d^2*m1^2*pp^2 + 90*d^3*m1^2*pp^2 - 
          24*d^4*m1^2*pp^2 - 48*pp^3 + 114*d*pp^3 - 67*d^2*pp^3 + 
          4*d^3*pp^3 + 2*d^4*pp^3)*TJI[d, pp, {{1, m1}, {1, m1}, {1, 0}}])/
      (12*(-2 + d)*(-1 + d)^2*(1 + d)*(-4 + 3*d)*(-2 + 3*d)*pp^2) - 
     ((-4*m1^2 + pp)*(72*d*m1^6 - 144*d^2*m1^6 + 88*d^3*m1^6 - 
          16*d^4*m1^6 + 96*m1^4*pp - 236*d*m1^4*pp + 122*d^2*m1^4*pp + 
          12*d^3*m1^4*pp - 10*d^4*m1^4*pp - 96*m1^2*pp^2 + 
          180*d*m1^2*pp^2 - 30*d^2*m1^2*pp^2 - 46*d^3*m1^2*pp^2 + 
          8*d^4*m1^2*pp^2 + 24*pp^3 - 54*d*pp^3 + 27*d^2*pp^3)*
        TJI[d, pp, {{2, m1}, {1, m1}, {1, 0}}])/
(12*(-3 + d)*(-2 + d)*(-1 + d)^2*(1 + d)*(-4 + 3*d)*(-2 + 3*d)*pp^2) /; (m1=!=0)&&(pp=!=0);

TVR[6 + d_Symbol, pp_, 
    {{1, m1_}, {1, m1_}, {4, 0}, {1, m1_}}] := 
   ((2304*d*m1^10 - 4608*d^2*m1^10 - 3520*d^3*m1^10 + 6976*d^4*m1^10 + 
          1472*d^5*m1^10 - 2624*d^6*m1^10 - 256*d^7*m1^10 + 
          256*d^8*m1^10 + 4608*m1^8*pp - 9600*d*m1^8*pp - 
          7680*d^2*m1^8*pp + 9312*d^3*m1^8*pp + 11680*d^4*m1^8*pp - 
          6560*d^5*m1^8*pp - 2880*d^6*m1^8*pp + 1088*d^7*m1^8*pp + 
          32*d^8*m1^8*pp + 6912*m1^6*pp^2 - 35712*d*m1^6*pp^2 + 
          1056*d^2*m1^6*pp^2 + 85296*d^3*m1^6*pp^2 - 
          37032*d^4*m1^6*pp^2 - 20484*d^5*m1^6*pp^2 + 
          8700*d^6*m1^6*pp^2 - 12*d^7*m1^6*pp^2 - 84*d^8*m1^6*pp^2 - 
          20736*m1^4*pp^3 + 58752*d*m1^4*pp^3 + 40560*d^2*m1^4*pp^3 - 
          150536*d^3*m1^4*pp^3 + 18980*d^4*m1^4*pp^3 + 
          48946*d^5*m1^4*pp^3 - 6748*d^6*m1^4*pp^3 - 146*d^7*m1^4*pp^3 + 
          128*d^8*m1^4*pp^3 + 3456*m1^2*pp^4 - 13824*d*m1^2*pp^4 - 
          4176*d^2*m1^2*pp^4 + 35784*d^3*m1^2*pp^4 - 
          8332*d^4*m1^2*pp^4 - 10162*d^5*m1^2*pp^4 + 
          1122*d^6*m1^2*pp^4 - 80*d^7*m1^2*pp^4 - 8*d^8*m1^2*pp^4 + 
          864*d*pp^5 - 216*d^2*pp^5 - 2268*d^3*pp^5 + 486*d^4*pp^5 + 
          729*d^5*pp^5)*TAI[d, 0, {{1, m1}}]^2)/
      (432*(-3 + d)*(-1 + d)^2*d^2*(1 + d)*(2 + d)*(3 + d)*(-4 + 3*d)*
        (-2 + 3*d)*(2 + 3*d)*m1^2*pp^3) + 
     ((4*m1^2 - pp)^3*TAI[d, 0, {{1, m1}}]*TBI[d, pp, {{1, m1}, {1, m1}}])/
      (16*(-3 + d)*(-1 + d)^2*d*(1 + d)*(3 + d)*m1^2) - 
     ((-1152*m1^10 + 2880*d*m1^10 + 320*d^2*m1^10 - 3648*d^3*m1^10 + 
          1088*d^4*m1^10 + 768*d^5*m1^10 - 256*d^6*m1^10 - 192*m1^8*pp + 
          1312*d*m1^8*pp + 2080*d^2*m1^8*pp - 6880*d^3*m1^8*pp + 
          4032*d^4*m1^8*pp - 192*d^5*m1^8*pp - 160*d^6*m1^8*pp + 
          6336*m1^6*pp^2 - 6960*d*m1^6*pp^2 - 16960*d^2*m1^6*pp^2 + 
          17292*d^3*m1^6*pp^2 - 1828*d^4*m1^6*pp^2 - 828*d^5*m1^6*pp^2 + 
          68*d^6*m1^6*pp^2 - 9504*m1^4*pp^3 + 3624*d*m1^4*pp^3 + 
          26104*d^2*m1^4*pp^3 - 10938*d^3*m1^4*pp^3 - 
          4976*d^4*m1^4*pp^3 + 330*d^5*m1^4*pp^3 + 400*d^6*m1^4*pp^3 + 
          6336*m1^2*pp^4 - 3640*d*m1^2*pp^4 - 14584*d^2*m1^2*pp^4 + 
          9190*d^3*m1^2*pp^4 + 1374*d^4*m1^2*pp^4 - 600*d^5*m1^2*pp^4 - 
          56*d^6*m1^2*pp^4 - 864*pp^5 + 732*d*pp^5 + 1864*d^2*pp^5 - 
          1695*d^3*pp^5 + 148*d^4*pp^5 + 36*d^5*pp^5 + 4*d^6*pp^5)*
        TJI[d, pp, {{1, m1}, {1, m1}, {1, 0}}])/
      (216*(-2 + d)*(-1 + d)^2*d*(1 + d)*(3 + d)*(-4 + 3*d)*(-2 + 3*d)*
        (2 + 3*d)*pp^3) + ((-4*m1^2 + pp)*
        (1152*m1^10 - 2880*d*m1^10 - 320*d^2*m1^10 + 3648*d^3*m1^10 - 
          1088*d^4*m1^10 - 768*d^5*m1^10 + 256*d^6*m1^10 + 576*m1^8*pp - 
          2208*d*m1^8*pp - 2304*d^2*m1^8*pp + 8000*d^3*m1^8*pp - 
          4256*d^4*m1^8*pp - 32*d^5*m1^8*pp + 224*d^6*m1^8*pp - 
          6336*m1^6*pp^2 + 6288*d*m1^6*pp^2 + 16320*d^2*m1^6*pp^2 - 
          14868*d^3*m1^6*pp^2 + 684*d^4*m1^6*pp^2 + 804*d^5*m1^6*pp^2 - 
          12*d^6*m1^6*pp^2 + 6912*m1^4*pp^3 - 3192*d*m1^4*pp^3 - 
          18160*d^2*m1^4*pp^3 + 7950*d^3*m1^4*pp^3 + 3128*d^4*m1^4*pp^3 - 
          78*d^5*m1^4*pp^3 - 160*d^6*m1^4*pp^3 - 3456*m1^2*pp^4 + 
          1800*d*m1^2*pp^4 + 8136*d^2*m1^2*pp^4 - 4322*d^3*m1^2*pp^4 - 
          1042*d^4*m1^2*pp^4 + 128*d^5*m1^2*pp^4 + 16*d^6*m1^2*pp^4 + 
          432*pp^5 - 324*d*pp^5 - 972*d^2*pp^5 + 729*d^3*pp^5)*
        TJI[d, pp, {{2, m1}, {1, m1}, {1, 0}}])/
      (216*(-3 + d)*(-2 + d)*(-1 + d)^2*d*(1 + d)*(3 + d)*(-4 + 3*d)*
        (-2 + 3*d)*(2 + 3*d)*pp^3) /; (m1=!=0)&&(pp=!=0);

TVR[6 + d_Symbol, pp_, {{2, m1_}, {1, m1_}, {3, 0}, {1, m1_}}] := 
   -((-1536*d*m1^10 + 2560*d^2*m1^10 + 3200*d^3*m1^10 - 3584*d^4*m1^10 - 
           2176*d^5*m1^10 + 1024*d^6*m1^10 + 512*d^7*m1^10 - 
           3072*m1^8*pp + 6528*d*m1^8*pp + 5376*d^2*m1^8*pp - 
           7520*d^3*m1^8*pp - 5568*d^4*m1^8*pp + 3232*d^5*m1^8*pp + 
           1344*d^6*m1^8*pp - 320*d^7*m1^8*pp - 2304*m1^6*pp^2 + 
           13824*d*m1^6*pp^2 + 192*d^2*m1^6*pp^2 - 31584*d^3*m1^6*pp^2 + 
           11232*d^4*m1^6*pp^2 + 8088*d^5*m1^6*pp^2 - 
           2304*d^6*m1^6*pp^2 - 24*d^7*m1^6*pp^2 + 6912*m1^4*pp^3 - 
           19584*d*m1^4*pp^3 - 12688*d^2*m1^4*pp^3 + 
           50584*d^3*m1^4*pp^3 - 7516*d^4*m1^4*pp^3 - 
           16982*d^5*m1^4*pp^3 + 2564*d^6*m1^4*pp^3 + 310*d^7*m1^4*pp^3 - 
           1152*m1^2*pp^4 + 4608*d*m1^2*pp^4 + 1392*d^2*m1^2*pp^4 - 
           11896*d^3*m1^2*pp^4 + 2820*d^4*m1^2*pp^4 + 
           3374*d^5*m1^2*pp^4 - 414*d^6*m1^2*pp^4 + 8*d^7*m1^2*pp^4 - 
           288*d*pp^5 + 72*d^2*pp^5 + 756*d^3*pp^5 - 162*d^4*pp^5 - 
           243*d^5*pp^5)*TAI[d, 0, {{1, m1}}]^2)/
      (144*(-1 + d)^2*d^2*(1 + d)*(2 + d)*(3 + d)*(-4 + 3*d)*(-2 + 3*d)*
        (2 + 3*d)*m1^2*pp^3) + 
     ((4*m1^2 - pp)^3*TAI[d, 0, {{1, m1}}]*TBI[d, pp, {{1, m1}, {1, m1}}])/
      (16*(-1 + d)^2*d*(1 + d)*(3 + d)*m1^2) - 
     ((2304*m1^10 - 5760*d*m1^10 - 640*d^2*m1^10 + 7296*d^3*m1^10 - 
          2176*d^4*m1^10 - 1536*d^5*m1^10 + 512*d^6*m1^10 - 
          1344*m1^8*pp + 1120*d*m1^8*pp - 1376*d^2*m1^8*pp + 
          5280*d^3*m1^8*pp - 4896*d^4*m1^8*pp + 1280*d^5*m1^8*pp - 
          64*d^6*m1^8*pp - 5184*m1^6*pp^2 + 7584*d*m1^6*pp^2 + 
          14800*d^2*m1^6*pp^2 - 21496*d^3*m1^6*pp^2 + 
          6120*d^4*m1^6*pp^2 + 280*d^5*m1^6*pp^2 - 184*d^6*m1^6*pp^2 + 
          9504*m1^4*pp^3 - 3720*d*m1^4*pp^3 - 27296*d^2*m1^4*pp^3 + 
          15378*d^3*m1^4*pp^3 + 4114*d^4*m1^4*pp^3 - 666*d^5*m1^4*pp^3 - 
          674*d^6*m1^4*pp^3 - 6336*m1^2*pp^4 + 5720*d*m1^2*pp^4 + 
          13712*d^2*m1^2*pp^4 - 13926*d^3*m1^2*pp^4 + 
          1284*d^4*m1^2*pp^4 + 946*d^5*m1^2*pp^4 - 80*d^6*m1^2*pp^4 + 
          864*pp^5 - 1020*d*pp^5 - 1636*d^2*pp^5 + 2311*d^3*pp^5 - 
          693*d^4*pp^5 + 20*d^5*pp^5 + 4*d^6*pp^5)*
        TJI[d, pp, {{1, m1}, {1, m1}, {1, 0}}])/
      (72*(-2 + d)*(-1 + d)^2*d*(1 + d)*(3 + d)*(-4 + 3*d)*(-2 + 3*d)*
        (2 + 3*d)*pp^3) + ((4*m1^2 - pp)*
        (-768*m1^10 + 1664*d*m1^10 + 768*d^2*m1^10 - 2176*d^3*m1^10 + 
          512*d^5*m1^10 + 192*m1^8*pp + 288*d*m1^8*pp + 704*d^2*m1^8*pp - 
          2272*d^3*m1^8*pp + 1024*d^4*m1^8*pp + 64*d^5*m1^8*pp + 
          1920*m1^6*pp^2 - 1824*d*m1^6*pp^2 - 5472*d^2*m1^6*pp^2 + 
          4584*d^3*m1^6*pp^2 - 168*d^5*m1^6*pp^2 - 2304*m1^4*pp^3 + 
          616*d*m1^4*pp^3 + 6192*d^2*m1^4*pp^3 - 1946*d^3*m1^4*pp^3 - 
          1128*d^4*m1^4*pp^3 - 230*d^5*m1^4*pp^3 + 1152*m1^2*pp^4 - 
          600*d*m1^2*pp^4 - 2744*d^2*m1^2*pp^4 + 1414*d^3*m1^2*pp^4 + 
          374*d^4*m1^2*pp^4 - 16*d^5*m1^2*pp^4 - 144*pp^5 + 108*d*pp^5 + 
          324*d^2*pp^5 - 243*d^3*pp^5)*
        TJI[d, pp, {{2, m1}, {1, m1}, {1, 0}}])/
      (72*(-2 + d)*(-1 + d)^2*d*(1 + d)*(3 + d)*(-4 + 3*d)*(-2 + 3*d)*
        (2 + 3*d)*pp^3) /; (m1=!=0)&&(pp=!=0);

five -> four

\(TFI[d_, \ 
        PP_ /; Head[PP] =!= List, {\ {0, \ m1_}, \ {n2_, \ 
            m2_}, \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {n3_, \ m3_}, \ {n4_, \ 
            m4_}, \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {n5_, \ m5_}\ }]\  := \ 

      TVI[d, PP, \ {{n5, m5}, \ {n2, m2}, \ {n3, m3}, \ {n4, m4}}]\)

\(TFI[
        d_, \ \ PP_ /; 
          Head[PP] =!= List, {\ {\((n2_)\), \ m2_}, {0, \ 
            m1_}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\((n4_)\), \ m4_}, \ {\((n3_)\), \ 
            m3_}, \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\((n5_)\), \ m5_}\ }]\  := \ 

      TVI[d, PP, \ {{n5, m5}, \ {n2, m2}, \ {n3, m3}, \ {n4, m4}}]\)

\(TFI[
        d_, \ \ \ PP_ /; 
          Head[PP] =!= List, {\ {\((n3_)\), \ m3_}, \ {\((n4_)\), \ 
            m4_}, 
		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {0, \ m1_}, \ {\((n2_)\), \ 
            m2_}, \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\((n5_)\), \ m5_}\ }]\  := \ 

      TVI[d, PP, \ {{n5, m5}, \ {n2, m2}, \ {n3, m3}, \ {n4, m4}}]\)

\(TFI[
        d_, \ \ \ PP_ /; 
          Head[PP] =!= List, {\ {\((n4_)\), \ m4_}, {\((n3_)\), \ 
            m3_}, 
		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\((n2_)\), \ m2_}, \ {0, \ 
            m1_}, \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\((n5_)\), \ m5_}\ }]\  := \ 

      TVI[d, PP, \ {{n5, m5}, \ {n2, m2}, \ {n3, m3}, \ {n4, m4}}]\)

TFI[d_,  PP_/;Head[PP]=!=List,{{(n1_), m1_}, 

    {(n2_), m2_}, 

    {(n3_), m3_}, 

    {(n4_), m4_}, {0, m5_}}] := 

  TBI[d, PP, {{n1, m1}, {n3, m3}}]*

   TBI[d, PP, {{n2, m2}, {n4, m4}}]

four -> three

\(TVI[d_, \ 
        PP_, {\ {0, \ m1_}, \ {n2_, \ 
            m2_}, \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {n3_, \ m3_}, \ {n4_, \ m4_}\ }]\  := \ 

      TAI[d, 0, {{n3, m3}}]\ TBI[d, PP, {{n2, m2}, {n4, m4}}]\)

\(TVI[d_, \ 
        PP_, {\ {n1_, \ m1_}, \ {n2_, \ 
            m2_}, \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {0, \ m3_}, \ {n4_, \ m4_}\ }]\  := \ 

      TAI[d, 0, {{n1, m1}}]\ TBI[d, PP, {{n2, m2}, {n4, m4}}]\)

TVI[d_, PP_, {{n1_, m1_}, {0, m2_}, {n3_, m3_}, {n4_, m4_}}] := 
  TJI[d, 0, {{n1, m1}, {n3, m3}, {n4, m4}}]

\(TVI[d_, \ 
        PP_, {\ {n1_, \ m1_}, \ {n2_, \ 
            m2_}, \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {n3_, \ m3_}, \ {0, \ m4_}\ }]\  := \ 

      TJI[d, PP, \ {{n2, m2}, {n1, m1}, {n3, m3}\ }]\)

three -> two

\(TJI[d_, \ 
        PP_, {\ {0, \ m1_}, \ {\((n2_)\), \ m2_}, \ {\((n3_)\), \ 
            m3_}\ }]\  := \ 

      TAI[d, 0, {{n2, m2}}]\ \ TAI[d, 0, {{n3, m3}}]\)

\(TJI[d_, \ 
        PP_, {\ {\((n1_)\), \ m1_}, \ {0, \ m2_}, \ {\((n3_)\), \ 
            m3_}\ }]\  := \ 

      TAI[d, 0, {{n1, m1}}]\ \ TAI[d, 0, {{n3, m3}}]\)

\(TJI[d_, \ 
        PP_, {\ {\((n1_)\), \ m1_}, \ {\((n2_)\), \ m2_}, \ {0, \ 
            m3_}\ }]\  := \ 

      TAI[d, 0, {{n1, m1}}]\ \ TAI[d, 0, {{n2, m2}}]\)

two -> one

\(\(\(\(TBI[d_, \ 
        PP_, {\ {0, \ m1_}, \ {\((n2_)\), \ 
            m2_}\ }]\)\(\ \)\)\(:=\)\(\ \)\(
\)\(TAI[d, 
        0, {{n2, m2}}]\)\(\ \)\)\)

\(\(\(\(TBI[d_, \ 
        PP_, {\ {\((n1_)\), \ m1_}, \ {0, \ 
            m2_}\ }]\)\(\ \)\)\(:=\)\(\ \)\(
\)\(TAI[d, 
        0, {{n1, m1}}]\)\(\ \)\)\)

TAI[d_, {{n_Integer, m_}}] := TAI[d, 0, {{n, m}}];

TAI[d_, {n_Integer, m_}] := TAI[d, 0, {{n, m}}];

TAI[d_, 0, {n_Integer, m_}] := TAI[d, 0, {{n, m}}];

TarcerRecurse and TFIRecurse

TFIRecurse & TFISimplify

This cancels scalar products.

$TLRComment = False;

TLRComment[s_String] := 
      WriteString["stdout", s, " "] /; $TLRComment === 
          True;

applytlrules[y_, fun_] :=fun[y /.TFI->TLR/.TLRules/.TLR->TFI];

TFIRecurse[z_, f_ : Identity]  :=  
      FixedPoint[ applytlrules[#, f] &, z, 1000];

applytlrules2[y_, fun_] :=fun[y /.TFI->TLR/.TLRules2/.TLR->TFI];

TFISimplify[z_, f_ : Identity]  :=  
      FixedPoint[ applytlrules2[#, f] &, z, 1000];

TFIRecurse[z_]:=z/.TFI->TLR//.TLRules/.TLR->TFI;

ExpandMaybe = Identity;

fN[li_List] := fN[li] = 
        li /. {0, _} :> {0, 0};

RM: new Feb. 1999  
need TLR[ ...]:= rule;
(copy from TFI definition) (in order to get e.g. 
TarcerRecurse[TFI[D,pp,{1,1,1,1,1},{{1,0},{1,M},{0,0},{1,m},{1,M}}]]
to work)

TLR[depp__, {a___, b_Integer, c___}] := 
      TLR[depp, {a, {b, 0}, c}];

tlrule[1]=TLR[d_, pp_, {0, 0, 0, 0, 0}, pr_]:>(NoTLRComment["tlrule1"];TFI[d, pp, 
                pr /. {0, _} :> {0, 0}]);

(* q1^2 *)

tlrule[2] = 
      TLR[dpp__, {v_?PQ, w_, x_, y_, z_}, {{n1_?PQ, m1_}, nm2_, nm3_, nm4_, 
            nm5_}] :> (TLRComment["tlrule2"]; 
          ExpandMaybe[
            If[v ≤ n1, 
              If[m1 === 0, 
                TLR[dpp, {0, w, x, y, z}, {{n1 - v, 0}, nm2, nm3, nm4, 
                    nm5}], (∑_(i = 0)^v) Binomial[v, i] TLR[
                      dpp,  {0, w, x, y, z}, {fN[{n1 - i, m1}], nm2, nm3, 
                        nm4, nm5}] ((((m1^2)))^(v - i))], 
              If[m1 === 0, 
                TLR[dpp, {v - n1, w, x, y, z}, {{0, 0}, nm2, nm3, nm4, 
                    nm5}], (∑_(i = 0)^n1) Binomial[n1, i] TLR[
                      dpp, {v - n1, w, x, y, z}, {fN[{n1 - i, m1}], nm2, nm3, 
                        nm4, nm5}] ((((m1^2)))^(n1 - i))]]]);

(Subscript[q, 2]^2)

tlrule[3] = 
      TLR[dpp__, {v_, w_?PQ, x_, y_, z_}, {nm1_, {n2_?PQ, m2_}, nm3_, nm4_, 
            nm5_}] :> (TLRComment["tlrule3"]; 
          ExpandMaybe[
            If[w ≤ n2, 
              If[m2 === 0, 
                TLR[dpp, {v, 0, x, y, z}, {nm1, {n2 - w, 0}, nm3, nm4, 
                    nm5}], (∑_(i = 0)^w) Binomial[w, i] TLR[
                      dpp, {v, 0, x, y, z}, {nm1, fN[{n2 - i, m2}], nm3, nm4, 
                        nm5}] ((((m2^2)))^(w - i))], 
              If[m2 === 0, 
                TLR[dpp, {v, w - n2, x, y, z}, {nm1, {0, 0}, nm3, nm4, 
                    nm5}], (∑_(i = 0)^n2) Binomial[n2, i] TLR[
                      dpp, {v, w - n2, x, y, z}, {nm1, fN[{n2 - i, m2}], nm3, 
                        nm4, nm5}] ((((m2^2)))^(n2 - i))]]]);

p·(Subscript[q, 1])

tlrule[4] = 
      TLR[dpp__, {v_, w_, x_?PQ, y_, z_}, {{n1_?PQ, m1_}, 
            nm2_, {n3_?PQ, m3_}, nm4_, 
            nm5_}] :> (TLRComment["tlrule4"]; 
          ExpandMaybe[
            (1/2) TLR[
                  dpp, {v + 1, w, x - 1, y, z}, {{n1, m1}, nm2, {n3, m3}, 
                    nm4, nm5}] - 
              (1/2) TLR[
                  dpp, {v, w, x - 1, y, z}, {{n1, m1}, nm2, fN[{n3 - 1, m3}], 
                    nm4, nm5}] + 
              (1/2) ({dpp}[[2]] - 
                    (m3^2)) TLR[
                  dpp, {v, w, x - 1, y, z}, {{n1, m1}, nm2, {n3, m3}, nm4, 
                    nm5}]]);

p·(Subscript[q, 2])

tlrule[5] = 
      TLR[dpp__, {v_, w_, x_, y_?PQ, z_}, {{n1_, m1_}, {n2_?PQ, m2_}, 
            nm3_, {n4_?PQ, m4_}, 
            nm5_}] :> (TLRComment["tlrule5"]; 
          ExpandMaybe[
            (1/2) TLR[
                  dpp, {v, w + 1, x, y - 1, z}, {{n1, m1}, {n2, m2}, 
                    nm3, {n4, m4}, nm5}] - 
              (1/2) TLR[
                  dpp, {v, w, x, y - 1, z}, {{n1, m1}, {n2, m2}, nm3, 
                    fN[{n4 - 1, m4}], nm5}] + 
              (1/2) ({dpp}[[2]] - 
                    (m4^2)) TLR[
                  dpp, {v, w, x, y - 1, z}, {{n1, m1}, {n2, m2}, 
                    nm3, {n4, m4}, nm5}]]);

(Subscript[q, 1])·(Subscript[q, 2])

tlrule[6] = 
      TLR[dpp__, {v_, w_, x_, y_, z_?PQ}, {{n1_, m1_}, {n2_, m2_}, nm3_, 
            nm4_, {n5_?PQ, 
              m5_}}] :> (TLRComment["tlrule6"]; 
          ExpandMaybe[
            (1/2) TLR[
                  dpp, {v + 1, w, x, y, z - 1}, {{n1, m1}, {n2, m2}, nm3, 
                    nm4, {n5, m5}}] - 
              (1/2) TLR[
                  dpp, {v, w, x, y, z - 1}, {{n1, m1}, {n2, m2}, nm3, nm4, 
                    fN[{n5 - 1, m5}]}] + 
              (1/2) TLR[
                  dpp, {v, w + 1, x, y, z - 1}, {{n1, m1}, {n2, m2}, nm3, 
                    nm4, {n5, m5}}] - 
              (1/2) (m5^2) TLR[
                  dpp, {v, w, x, y, z - 1}, {{n1, m1}, {n2, m2}, nm3, 
                    nm4, {n5, m5}}]]);

(* q1.q1  --> q1.p *)

tlrule[7] = 
      TLR[d_, pp_, 
          dp___, {v_?PQ, w_, x_, y_, z_}, {{0, _}, nm2_, {n3_?PQ, m3_}, nm4_, 
            nm5_}] :> (TLRComment["tlrule7"]; 
          ExpandMaybe[
            TLR[d, pp, 
                dp, {v - 1, w, x, y, z}, {{0, 0}, nm2, fN[{n3 - 1, m3}], nm4, 
                  nm5}] + 
              2 TLR[d, pp, 
                  dp, {v - 1, w, x + 1, y, z}, {{0, 0}, nm2, {n3, m3}, nm4, 
                    nm5}] - (pp - (m3^2)) TLR[d, pp, 
                  dp, {v - 1, w, x, y, z}, {{0, 0}, nm2, {n3, m3}, nm4, 
                    nm5}]]);

(* q2.q2  --> q2.p *)

tlrule[8] = 
      TLR[d_, pp_, 
          dp___, {v_, w_?PQ, x_, y_, z_}, {nm1_, {0, _}, nm3_, {n4_?PQ, m4_}, 
            nm5_}] :> (TLRComment["tlrule8"]; 
          ExpandMaybe[
            TLR[d, pp, 
                dp, {v, w - 1, x, y, z}, {nm1, {0, 0}, nm3, fN[{n4 - 1, m4}], 
                  nm5}] + 
              2 TLR[d, pp, 
                  dp, {v, w - 1, x, y + 1, z}, {nm1, {0, 0}, nm3, {n4, m4}, 
                    nm5}] - (pp - (m4^2)) TLR[d, pp, 
                  dp, {v, w - 1, x, y, z}, {nm1, {0, 0}, nm3, {n4, m4}, 
                    nm5}]]);

(* eq. (3) - (5) *)

\(tlrule[9] = 
      TLR[d_, pp_, {v_, w_, x_, y_, z_?PQ}, {nm1_, nm2_, nm3_, 
            nm4_, {0, _}}] :> \((TLRComment["tlrule9"]; 
          FunctionExpand[
            ExpandMaybe[∑\+\(k = 0\)\%\(Floor[z\/2]\)\(∑\+\(i = 0\)\%k\(∑\+\(j = 0\)\%k\((Gamma[k + 1\/2]\ Gamma[d\/2 - 1\/2]\ Binomial[z, 
                            2\ k]\ Binomial[k, i]\ Binomial[k, 
                            j]\ \((\(-1\))\)\^\(i + j\)\ pp\^\(i + j - z\)\ TLR[d, pp, {v + i, w + j, x + z - 2\ i, y + z - 2\ j, 0}, {nm1, nm2, nm3, 
                              nm4, {0, 0}}])\)/\((Gamma[1\/2]\ Gamma[
                            k + d\/2 - 1\/2])\)\)\)]])\); \)

OLD trule[10]:

tlrule[10] = 
(* (9) and (10) *)
TLR[d_, pp_, {v_,w_,x_?PQ,y_?PQ,z_}, {{n1_?PQ,m1_},{n2_?PQ,m2_}, 
                                {0,_},{0,_},nm5_}] :>  
( TLRComment["tlrule10"]; 0 /; OddQ[x+y] );

New tlrule[10]:

tlrule[10] = 
   TLR[d_, pp_, {v_, w_, (x_)?PNQ, (y_)?PNQ, z_}, 
     {{(n1_)?PQ, m1_}, {(n2_)?PQ, m2_}, {0, _}, {0, _}, nm5_}] :> (TLRComment["tlrule10"]; 0 /; OddQ[x + y]);

Bug in  tlrule[11] and tlrule[12] fixed on Feb. 26th 1998 by R.Scharf 
with thanks to York Schroeder for locating the bug.

tlrule[11] = 
TLR[d_, pp_, {v_,w_,x_?PQ,y_?PQ,z_}, {nm1_,nm2_,{0,_},{0,_},{(n5_)?PQ, m5_}}] :> 
(
TLRComment["tlrule11"];
 ExpandMaybe[
 pp/(d+x+y-2)( (x-1) TLR[d, pp, {v+1,w,x-2,y,z}, 
                         {{n1,m1},{n2,m2},{0,0},{0,0},{n5,m5}}] +
                   y TLR[d, pp, {v,w,x-1,y-1,z+1}, 
                         {nm1,nm2,{0,0},{0,0},{n5,m5}}]
             )
           ] /; EvenQ[x+y] && x > 1
);

tlrule[12] = 
TLR[d_, pp_, {v_,w_,x_?PQ,y_?PQ,z_}, {nm1_,nm2_,{0,_},{0,_},{(n5_)?PQ, m5_}}] :> 
(
TLRComment["tlrule12"];
 ExpandMaybe[
 pp/(d+x+y-2)( (y-1) TLR[d, pp, {v,w+1,x,y-2,z}, 
                         {{n1,m1},{n2,m2},{0,0},{0,0},{n5,m5}}] +
                   x TLR[d, pp, {v,w,x-1,y-1,z+1}, 
                         {nm1,nm2,{0,0},{0,0},{n5,m5}}]
             )
           ] /; EvenQ[x+y] && y > 1
);

(*symm.   1 <-> 3  &&  2 <-> 4 *)

\(tlrule[13] = 
      TLR[d_, pp_, {0, 0, 0, y_?PQ, 0}, {nm1_, {0, 0}, nm3_, {n4_?PQ, m4_}, 
            nm5_}] :> \((TLRComment["tlrule13"]; ∑\+\(i = 0\)\%y Binomial[y, i]\ pp\^\(y - i\)\ \((\(-1\))\)\^i\ TLR[d, 
                pp, {0, 0, 0, i, 0}, {nm3, {n4, m4}, nm1, {0, 0}, 
                  nm5}])\); \)

(*symm.   1 <-> 3  &&  2 <-> 4 *)

\(tlrule[14] = 
      TLR[d_, pp_, {0, 0, x_?PQ, 0, 0}, {{0, 0}, nm2_, {n3_?PQ, m3_}, nm4_, 
            nm5_}] :> \((TLRComment["tlrule14"]; ∑\+\(i = 0\)\%x Binomial[x, i]\ pp\^\(x - i\)\ \((\(-1\))\)\^i\ TLR[d, 
                pp, {0, 0, i, 0, 0}, {{n3, m3}, nm4, {0, 0}, nm2, 
                  nm5}])\); \)

(* 0's *)

\(tlrule[15] = 
      HoldPattern[
          TLR[d_, pp_, {v_, w_, x_, y_, 
              z_}, {{n1_, 0}, {n2_, 
                0}, {0, _}, {0, _}, {0, _}}]] :> \((TLRComment["tlrule15"]; 0)\); \)

\(tlrule[16] = 
      TLR[d_, pp_, {0, 0, x_?PQ, y_?PQ, 
            0}, {{n1_, _}, {n2_, _}, {0, _}, {0, _}, {0, _}}] :> \((TLRComment["tlrule16"]; 0 /; OddQ[x] || OddQ[y])\); \)

bug in tlrule[17] fixedApril 23rd 1999

\(tlrule[17] = 
      TLR[d_, pp_, {a_?PQ, b_, 0, de_, 
            0}, {{0, _}, {n2_, m2_}, {0, _}, {n4_, m4_}, {n5_, 
              m5_}}] :> \((TLRComment["tlrule17"]; ∑\+\(i = 0\)\%a\(∑\+\(j = 0\)\%i Binomial[a, i]\ Binomial[i, 
                  j]\ \((1 + \((\(-1\))\)\^j)\)\ 2\^\(j - 1\)\ TLR[d, 
                  pp, {a - i, b + i - j, 0, de, 
                    j}, {{n5, m5}, {n2, m2}, {0, 0}, {n4, m4}, {0, 
                      0}}]\))\); \)

\(tlrule[18] = 
      TLR[d_, pp_, 
          dp_, {a_?PNQ, b_?PNQ}, {1, 0, 0, 0, 
            0}, {{0, 0}, {n2_, m2_}, {n3_, m3_}, {n4_, m4_}, {n5_, 
              m5_}}] :> \((TLRComment["tlrule18"]; ∑\+\(i = 0\)\%a\(∑\+\(j = 0\)\%b Binomial[a, i]\ Binomial[b, 
                  j]\ \((\((pp + m3\^2)\)\ TLR[d, pp, 
                        dp, {i, 
                          j}, {{n3, m3}, {n4, m4}, {0, 0}, {n2, m2}, {n5, 
                            m5}}] + 
                    TLR[d, pp, 
                      dp, {i, 
                        j}, {{n3 - 1, m3}, {n4, m4}, {0, 0}, {n2, m2}, {n5, 
                          m5}}] - 
                    2\ TLR[d, pp, 
                        dp, {i, j}, {0, 0, 1, 0, 
                          0}, {{n3, m3}, {n4, m4}, {0, 0}, {n2, m2}, {n5, 
                            m5}}])\)\ \((\(-1\))\)\^\(i + j\)\ dp\^\(a + b - i - j\)\))\); \)

\(tlrule[19] = 
      TLR[d_, pp_, 
          dp_, {a_?PNQ, b_?PNQ}, {0, 1, 0, 0, 
            0}, {{n1_, m1_}, {0, 0}, {n3_, m3_}, {n4_, m4_}, {n5_, 
              m5_}}] :> \((TLRComment["tlrule18"]; ∑\+\(i = 0\)\%a\(∑\+\(j = 0\)\%b Binomial[a, i]\ Binomial[b, 
                  j]\ \((\((pp + m4\^2)\)\ TLR[d, pp, 
                        dp, {i, 
                          j}, {{n3, m3}, {n4, m4}, {n1, m1}, {0, 0}, {n5, 
                            m5}}] + 
                    TLR[d, pp, 
                      dp, {i, 
                        j}, {{n3, m3}, {n4 - 1, m4}, {n1, m1}, {0, 0}, {n5, 
                          m5}}] - 
                    2\ TLR[d, pp, 
                        dp, {i, j}, {0, 0, 0, 1, 
                          0}, {{n3, m3}, {n4, m4}, {n1, m1}, {0, 0}, {n5, 
                            m5}}])\)\ \((\(-1\))\)\^\(i + j\)\ dp\^\(a + b - i - j\)\))\); \)

(* q1.q1  --> q1.p *)

\(tlrule[20] = 
      TLR[d_, pp_, {v_?PQ, w_, x_, y_, 0}, {{0, _}, nm2_, {0, _}, 
            nm4_, {n5_?PQ, 
              m5_}}] :> \((TLRComment["tlrule20"]; ∑\+\(j = 0\)\%x\(∑\+\(i = 0\)\%v\(∑\+\(r = 0\)\%i Binomial[i, 
                    r]\ Binomial[v, i]\ Binomial[x, j]\ 2\^r\ TLR[d, 
                    pp, {v - i, \(-r\) + i + w, j, x - j + y, r}, {{n5, m5}, 
                      nm2, 0, nm4, 0}]\)\))\); \)

tlrule[21] = 
      TLR[d_, pp_, {v_, w_?PQ, x_, y_, 0}, {nm1_, {0, _}, nm3_, {0, _}, 
            nm5_}] :> (TLRComment["tlrule21"]; 
          TLR[d, pp, {w, v, y, x, 0}, {{0, 0}, nm1, {0, 0}, nm3, 
              nm5}]);

\(tlrule[22] = 
      TLR[d_, pp_, 
          dp_, {0, b_?PNQ}, {0, w_?PNQ, 0, y_?PNQ, 1}, {nm1_, nm2_, nm3_, 
            nm4_, {0, 
              0}}] :> \((TLRComment["tlrule22"]; \(1\/pp\) \((TLR[d, pp, dp, {0, b}, {0, w, 1, y + 1, 0}, {nm1, nm2, nm3, 
                  nm4, {0, 0}}])\))\); \)

Using  FeynCalc : 

TIDL[{{k1, μ}, {k1, ν}}, {p}]
(((g^(μν)) ((((k1·p))^2)-(k1^2) (p^2)))/((1 - D) (p^2)))-(((p^μ) (p^ν) (D (((k1·p))^2)-(k1^2) (p^2)))/((1 - D) (p^4)))
Collect2[Calc[ FVD[OPEDelta, μ] FVD[k2, ν] %], {k1, k2}]
((k2·Δ (((k1·p))^2))/((1 - D) (p^2)))-((D k2·p Δ·p (((k1·p))^2))/((1 - D) (p^4)))-(((k1^2) k2·Δ)/(1 - D))+(((k1^2) k2·p Δ·p)/((1 - D) (p^2)))

\(tlrule[23] = 
      TLR[d_, pp_, 
          dp_, {1, b_?PNQ}, {0, w_?PNQ, 0, y_?PNQ, 1}, {nm1_, nm2_, nm3_, 
            nm4_, {0, 
              0}}] :> \((TLRComment["tlrule23"]; \(1\/\(\((1 - d)\) pp\)\) \((TLR[d, pp, 
                  dp, {0, b + 1}, {0, w, 2, y, 0}, {nm1, nm2, nm3, 
                    nm4, {0, 0}}])\) - \(\(d\ \ dp\)\/\(\((1 - d)\) 
                    pp\^2\)\) \((TLR[d, pp, 
                  dp, {0, b}, {0, w, 2, y + 1, 0}, {nm1, nm2, nm3, 
                    nm4, {0, 0}}])\) - \(1\/\(1 - d\)\) 
              TLR[d, pp, 
                dp, {0, b + 1}, {1, w, 0, y, 0}, {nm1, nm2, nm3, 
                  nm4, {0, 0}}] + \(dp\/\(\((1 - d)\) pp\)\) 
              TLR[d, pp, 
                dp, {0, b}, {1, w, 0, y + 1, 0}, {nm1, nm2, nm3, 
                  nm4, {0, 0}}])\); \)

TIDL[{{k1, μ}, {k1, ν}, {k1, ρ}}, {p}]
-(((p^μ) (p^ν) (p^ρ) k1·p ((D + 2) (((k1·p))^2)-3 (k1^2) (p^2)))/((1 - D) (p^6)))+(((p^μ) (g^(νρ)) k1·p ((((k1·p))^2)-(k1^2) (p^2)))/((1 - D) (p^4)))+(((g^(μρ)) (p^ν) k1·p ((((k1·p))^2)-(k1^2) (p^2)))/((1 - D) (p^4)))+(((g^(μν)) (p^ρ) k1·p ((((k1·p))^2)-(k1^2) (p^2)))/((1 - D) (p^4)))
Collect2[
        Calc[ FVD[OPEDelta, μ] FVD[OPEDelta, ν] 
            FVD[k2, ρ] %], {k1, k2}]
((2 k2·Δ Δ·p (((k1·p))^3))/((1 - D) (p^4)))-(((D + 2) k2·p (((Δ·p))^2) (((k1·p))^3))/((1 - D) (p^6)))-((2 (k1^2) k2·Δ Δ·p k1·p)/((1 - D) (p^2)))+((3 (k1^2) k2·p (((Δ·p))^2) k1·p)/((1 - D) (p^4)))

tlrule[24] = 
      TLR[d_, pp_, 
          dp_, {2, b_?PNQ}, {0, w_?PNQ, 0, y_?PNQ, 1}, {nm1_, nm2_, nm3_, 
            nm4_, {0, 0}}] :> (TLRComment["tlrule24"]; 
          2  dp  (1/((1 - d) (pp^2))) (TLR[d, pp, 
                    dp, {0, b + 1}, {0, w, 3, y, 0}, {nm1, nm2, nm3, 
                      nm4, {0, 0}}]) - (((d + 
                        2)  (dp^2))/((1 - d) (pp^3))) (TLR[d, 
                  pp, dp, {0, b}, {0, w, 3, y + 1, 0}, {nm1, nm2, nm3, 
                    nm4, {0, 0}}]) - ((2  dp)/((1 - d) pp)) 
              TLR[d, pp, 
                dp, {0, b + 1}, {1, w, 1, y, 0}, {nm1, nm2, nm3, 
                  nm4, {0, 0}}] + ((3  (dp^2))/((1 - d) (pp^2))) 
              TLR[d, pp, 
                dp, {0, b}, {1, w, 1, y + 1, 0}, {nm1, nm2, nm3, 
                  nm4, {0, 0}}]);

Symmetry 1 <--> 2; added February 21 1999 by R.M.

\(\(\(tlrule[25] = 
      TLR[dim_, 
          pp_, \ {a_, \ b_, \ \((c_)\)?PNQ, \ \((d_)\)?PQ, \ 
            0}, \ \(\({nm1_, \ nm2_, \ nm3_, \ {0, \ _}, \ 
            nm5_}\)\(
\)\)\ \ \ ]\  :> \((TLRComment["tlrule25"]; \ 
            TLR[dim, \ 
              pp, \ {b, a, \ d, \ c, \ 0}, \ {nm2, \ nm1, \ {0, 0}, \ nm3, \ 
                nm5}])\)\  /; \ d\  > \ c; \)\(
\)\)\)

TLRules = Array[tlrule, 25];

TLRules2 = Drop[TLRules, {9, 9}];

TLR[__, {{_, _}, {0, _}, {_, _}, {0, _}, {_, 0}}] := 0;

TLR[__, {{0, _}, {_, _}, {0, _}, {_, _}, {_, 0}}] := 0;

TLR[__, {{0, _}, {_, _}, {_, 0}, {_, _}, {0, _}}] := 0;

TLR[__, {{_, 0}, {_, _}, {0, _}, {_, _}, {0, _}}] := 0;

TLR[__, {{_, _}, {_, 0}, {_, _}, {0, _}, {0, _}}] := 0;

TLR[__, {{_, _}, {0, _}, {_, _}, {_, 0}, {0, _}}] := 0;

SetAttributes[seeetDelayed, HoldAll];

seeetDelayed[TLR[a__], b_] := SetDelayed @@ {TLR[a], b}

lis1 = 
      Permutations[{{0, _}, {0, _}, {0, _}, {_, _}, {_, 0}}];

lis2 = 
      Table[TLR[__, 
              lis1[[i]]] :> 
            0, {i, Length[lis1]}] /. 
        RuleDelayed -> seeetDelayed;

Clear[lis1, lis2];

T*IS

These functions below are never used.
They can serve as tests.

\(TJIS[d_, 
        Power[M_, \ 
          2], \ {{α_, 0}, {β_, 0}, {γ_, 
            0}}] := \(\(\(SMu\^2\) \((\(\((\(-1\))\)\^\(α + β + γ + 1\)\/\((\(-\ M\^2\))\)\^\(α + β + γ - 4\)\) 
            Exp[\(-I\)\ Pi\ \((d - 
                    4)\)]\ \(Gamma[α + β + γ - 
                    d]\/Gamma[\(3\/2\) 
                      d - α - β - γ]\) \(Gamma[
                  d\/2 - α]\/Gamma[α]\) \(Gamma[
                  d\/2 - β]\/Gamma[β]\) 
            Gamma[d\/2 - γ]\/Gamma[γ])\)\)\(
\)\)\)

\(TJIS[d_, 
        Power[M_, \ 
          2], \ {{β_, 0}, {γ_, 0}, {α_, M_}}] := 
	
      TJIS[d, M^2, {{α, M}, {β, 0}, {γ, 0}}]\)

\(TJIS[d_, 
        Power[M_, \ 
          2], \ {{β_, 0}, {α_, M_}, {γ_, 0}}] := 
	
      TJIS[d, M^2, {{α, M}, {β, 0}, {γ, 0}}]\)

\(TJIS[d_, 
        Power[M_, \ 
          2], \ {{α_, M_}, {β_, 0}, {γ_, 
            0}}] := 
\(SMu\^2\/\((M\^2)\)\^\(α + β + γ - 4\)\) \(\((\(-1\))\)\^\(α + β + γ + 
              1\)\) \(Gamma[α + β + γ - 
                d]\/\(Gamma[α] Gamma[β] 
              Gamma[γ]\)\) \(\(Gamma[d\/2 - γ] 
              Gamma[d\/2 - β]\)\/Gamma[
              d - β - γ]\) \(Gamma[β + γ - d\/2] Gamma[2\ d - α - 2  β - 2  γ]\)\/Gamma[\(3  d\)\/2 - α - β - γ]\)

TJIS[d_, 0,  {{β_, 0}, {α_, M_}, {γ_, 0}}] := 
      TJIS[d, 0, {{α, M}, {β, 0}, {γ, 0}}]

TJIS[d_, 0,  {{β_, 0}, {γ_, 0}, {α_, M_}}] := 
      TJIS[d, 0, {{α, M}, {β, 0}, {γ, 0}}]

\(TJIS[d_, 
        0, \ {{α_, M_}, {β_, 0}, {γ_, 
            0}}] := 
\(SMu\^2\/\((M\^2)\)\^\(α + β + γ - 4\)\) \(\((\(-1\))\)\^\(α + β + γ + 
              1\)\) \(Gamma[α + β + γ - 
                d]\/\(Gamma[α] Gamma[β] 
              Gamma[γ]\)\) \((Gamma[β + γ - d/2] 
              Gamma[d/2 - β] Gamma[d/2 - γ])\)/
          Gamma[d/2]\)

TJIS[d_, 0,  {{β_, M_}, {α_, 0}, {γ_, M_}}] := 
      TJIS[d, 0, {{α, 0}, {β, M}, {γ, M}}]

TJIS[d_, 0,  {{β_, M_}, {γ_, M_}, {α_, 0}}] := 
      TJIS[d, 0, {{α, 0}, {β, M}, {γ, M}}]

\(TJIS[d_, 
        0, \ {{α_, 0}, {β_, M_}, {γ_, 
            M_}}] := 
\(SMu\^2\/\((M\^2)\)\^\(α + β + γ - 4\)\) \(\((\(-1\))\)\^\(α + β + γ + 
              1\)\) \(Gamma[α + β + γ - 
                d]\/\(Gamma[β] Gamma[γ]\)\) \(Gamma[
              d\/2 - α]\/Gamma[
              d\/2]\) \(Gamma[α + γ - d\/2] Gamma[α + β - d\/2]\)\/Gamma[2  α + β + γ - d]\)

TJIS[d_, 
        0, {{α_, 0}, {β_, M_}, {γ_, 
            M_}}] := ((SMu^2) (((-1))^(α + β + γ + 1)) Gamma[α + β + γ - d] Gamma[
              (d/2) - α] (Gamma[α + γ - 
                    (d/2)] Gamma[α + β - 
                    (d/2)]))/(((((M^2)))^(α + β + γ - 4)) (Gamma[β] Gamma[γ]) Gamma[(d/2)] Gamma[
              2 α + β + γ - d])

(2.65)

TAIS[d_, 
        0, {{1, M_}}] :=  -I  (M^2)  SMu Gamma[(4 - d)/2 - 
            1]

TBIS[d_, 0, {{α_, 0}, {β_, 0}}] := 0;

\(TBIS[d_, 
        Power[M_, \ 
          2], {{α_, 0}, {β_, 0}}] := \(\(\(\(I\)\(\ \)\) 
        SMu\ *\((\(\((\(-1\))\)\^\(α + β\)\/\((\(-\ M\^2\))\)\^\(α + β - 2\)\) 
            Exp[\(-I\)\ Pi\ \((d - 4)\)/
                  2]\ \(Gamma[α + β - d/2]\/\(Gamma[α] 
                  Gamma[β]\)\) \(Gamma[d/2 - α] Gamma[d/2 - β]\)\/Gamma[d - α - β])\)\)\(
\)\)\)

\(TJIS[d_Symbol, M_\^2, {{1, M_}, {1, M_}, {1, M_}}] := 
      M\^2\ SMu\^2\ \((\(-\(6\/\((\(-4\) + d)\)\^2\)\) + 
            17\/\(2\ \((\(-4\) + d)\)\) + 
            1\/8\ \((\(-59\) - 
                  2\ π\^2)\) + \((\(-4\) + d)\)\^3\ TJI111e + \(1\/1920\) \((\((\(-4\) + d)\)\^2\ \((16755 - 4750\ π\^2 - 14\ π\^4 + 
                      3840\ π\^2\ Log[2] - 12080\ Zeta[3])\))\) + 
            1\/96\ \((\(-4\) + d)\)\ \((195 + 98\ π\^2 - 
                  48\ Zeta[3])\))\)\)

TComment

$CommentNotebook = True;

\(TComment[s_String, b_] := 
      If[If[\(\(\)\(! \((ValueQ[$TarcerRecursed])\)\)\), $TarcerRecursed = {}]; \(\(\)\(! \((MemberQ[$TarcerRecursed, {s, b}])\)\)\), AppendTo[$TarcerRecursed, {s, b}]; 
          If[$Notebooks && $CommentNotebook === True, 
            If[$commentnb === False, $commentnb = True; $cnb = 
                NotebookCreate[
                  StyleDefinitions -> "NaturalColor.nb", 
                  Magnification -> 1.5]; 
              SetOptions[$cnb, 
                WindowMargins -> {{0, Automatic}, {Automatic, 0}}]; 
              SetOptions[$cnb, WindowSize -> {500, 300}]; 
              NotebookWrite[$cnb, 
                Cell[BoxData[
                    ToBoxes[b /. TFIC -> TFI, 
                      StandardForm]], "Input", 
                  CellDingbat -> s <> " "]], 
              NotebookWrite[$cnb, 
                Cell[BoxData[
                    ToBoxes[b /. TFIC -> TFI, 
                      StandardForm]], "Input", 
                  CellDingbat -> s <> " "]]], 
            Print[s, 
              b /. {TFIC :> TFI, TVIC :> TVI, 
                  TJIC :> TJI, TBIC :> TBI, 
                  TAIC :> TAI}]]] /; $Comment === True\)

$TTable

$TTable = {};

TarcerRecurse

FEPrint[a__String] := Print[a]  /;  $Notebooks === False;

\(FEPrint[a__String] := 
      If[Head[$cnb] === NotebookObject, 
		NotebookWrite[$cnb, 
            Cell[StringJoin @@ {a}, "SmallText"]]; 
	
          SelectionMove[$cnb, After, Cell]] /; $Notebooks === 
          True; \)

Options[TarcerRecurse] = {Factor -> Factor, 
        TimeConstraint -> Infinity, Table :> $TTable, Together -> Together, 
        Replace -> {}};

TarcerRecurse[z_, opts___]:=Catch[Block[{basisdone, factorfun, fertig, dummm1, dummm2, getvars, 
                te, time, cutoff, tog, ttable, facfun, nexp, oldexpr, rec, 
                special, sti, tl, tcheck, tjrd, tvrd, together, 
                vars, $cc = 0, $cv},
			time  =  AbsoluteTime[];
			ttable = Table /. {opts} /. 
                    Options[TarcerRecurse];
			If[! MatchQ[ttable, {__RuleDelayed}], 
                  ttable = {}];
			special = Replace /. {opts} /. 
                    Options[TarcerRecurse];
			cutoff = TimeConstraint /. {opts} /. 
                    Options[TarcerRecurse];
			together = Together /. {opts} /. 
                    Options[TarcerRecurse];factorfun = Factor /. {opts} /. 
                    Options[TarcerRecurse];$commentnb = False;\(tog = \((\((If[#1 > 1001, 
                              If[$Comment, 
                                FEPrint["coefficient # " <> 
                                    If[ValueQ[$cv], ToString[$cc], ""] <> 
                                    If[ValueQ[$cv], " (" <> 
                                        ToString[$cv] <> ")", ""] <> " simplified, ", ToString[#1]]]]; #2)\) &)\)[LeafCount[#1], \($cc++\); 
                      together[#1]] &\);If[! (ValueQ[$TarcerRecursed]), $TarcerRecursed = {}];$TarcerRecursed = {};If[\(! \((MemberQ[$ContextPath, "HighEnergyPhysics`fcloops`ToTFi`"])\)\),t0 = z,t0=HighEnergyPhysics`fcloops`ToTFi`ToTFi[z]/.{HighEnergyPhysics`fcloops`TFi`TFi:>TFI,HighEnergyPhysics`fcloops`ToTFi`ToTFi :> Identity}];
			If[ttable =!= {}, t0 = t0 /. ttable];
			getvars=Select[Variables[#1],MatchQ[#1,TFI[__]|TVI[__]|TJI[__]|TBI[__]|TAI[__]]&]&;
			If[! FreeQ[t0, TFI], 
                  t0 = Collect[t0 = TFIRecurse[t0] /. ttable, getvars[t0], 
                      tog]];
			\(If[\(! FreeQ[t0, 
                      TVI[_Plus, __]]\), 
			tvrd = 
                    Select[DownValues[
                          TVR], \(! FreeQ[#, TVI[_ - 2, __]]\) &] /. 
                      TVR -> TVI; 
				t0 = 
                    Collect[
                      t0 = \(\(\(t0 /. tvrd\) /. tvrd\) /. tvrd\) /. tvrd, 
                      getvars[t0], 
                      tog]; 
				If[\(! FreeQ[t0, TVI[_Plus, __]]\), 	
                    t0 = Collect[t0 = t0 //. tvrd, getvars[t0], tog]]]\);
				
			tjrd = 
                  Select[DownValues[TJR], ! FreeQ[#, TJI[_ - 2, __]] &] /. 
                    TJR -> TJI;
			
			\(If[\(! FreeQ[t0, 
                      TJI[_Plus, __]]\), 
			tjrd = 
                    Select[DownValues[
                          TJR], \(! FreeQ[#, TJI[_ - 2, __]]\) &] /. 
                      TJR -> TJI; 
				t0 = 
                    Collect[
                      t0 = \(\(\(t0 /. tjrd\) /. tjrd\) /. tjrd\) /. tjrd, 
                      getvars[t0], 
                      tog]; 
				If[\(! FreeQ[t0, TJI[_Plus, __]]\), 	
                    t0 = Collect[t0 = t0 //. tjrd, getvars[t0], tog]]]\);
			
			\(rec[expr_, {ti_, tr_}] := \((sti = ToString[ti]; 
                    tcheck = \(\(\)\(! \((FreeQ[expr, 
                            ti])\)\)\); oldexpr =  . ; nexp = expr; 
                    While[tcheck === True && oldexpr =!= nexp, 
                      oldexpr = nexp; 
                      If[$Comment, \((If[#1 === 1, 
                                FEPrint[
                                  sti <> " level, there is 1 " <> sti], 
                                FEPrint[sti <> " level, there are ", 
                                  ToString[#1], " " <> 
                                    sti <> "'s "]] &)\)[
                          Count[nexp, 
                            ti[__], \(-1\)]]]; 
				
						nexp = \(nexp /. ti -> tr\) /. 
                          tr -> ti; 
							If[\(! FreeQ[nexp, 
                            TJI[_Plus, __]]\), 
                        nexp = 
                          nexp //. tjrd]; 
						
						tvars = 
                        getvars[nexp]; $cc = 0; $cv = Length[tvars]; 
                      nexp = Collect[nexp, tvars, tog]; 
						If[
                        cutuff =!= Infinity, 
                        If[AbsoluteTime[] - time > cutoff, 
                          Throw[$Failed]]]; 
						
						tcheck = \(! \((FreeQ[nexp, ti])\)\); ]; nexp)\)\);basisdone = 0;tl={{TFI,TFR},{TVI,TVR},{TJI, TJR},{TBI, TBR},{TAI, TAR}};Do[
                  t0 = rec[t0, tl[[i]]]; 
                  If[Head[t0] === Plus, 
                    fertig = 
                      Select[t0, ! (FreeQ[#1, 
                              tl[[i, 
                                1]]])]; 
                    basisdone = basisdone + fertig; t0 = t0 - fertig], {i, 
                    Length[tl]}];t0 = t0 + basisdone;$cv =  . ;t0 = 
                  Collect[
                    t0, {TAI[__], TAI[a1__] TAI[a2__], TAI[__], TBI[__], 
                      TJI[__], TVI[__], TFI[__]}, factorfun];If[$Comment, 
                  FEPrint["the result has ", 
                    ToString[
                      Length[t0 + dummm1 + dummm2] - 2], " terms"]];If[$Notebooks, 
                  SetSelectedNotebook[EvaluationNotebook[]]];t0]];

TarcerExpand

\(Options[TarcerExpand] = {
		TarcerReduce :> $BasisIntegrals, 
	
        TarcerRecurse -> False, Zeta -> True}; \)

TarcerExpand[expr_, d_Symbol -> fe_ /; Length[Variables[fe]] === 1, 
        ru___Rule] := TarcerExpand[expr, d  -> fe, 0, ru];

TarcerExpand[expr_, 
        d_Symbol -> fe_ /; Length[Variables[fe]] === 1, 
        order_Integer?NonNegative, opts___Rule]:=Block[{},Catch[e = First[Variables[fe]];{ints, recurse, 
                  zeta2replace} = {TarcerReduce, TarcerRecurse, Zeta} /. 
                  Flatten[{opts, Options[TarcerExpand]}];\(If[
                zeta2replace =!= True && 
                  MemberQ[$ContextPath, "HighEnergyPhysics`general`Zeta2`"], z2rule = {}, 
                z2rule = π\^2 :> 
                    6\ ToExpression["HighEnergyPhysics`general`Zeta2`Zeta2"]]\);ints = Flatten[ints];If[recurse =!= True, ta0 = expr, 
                ta0 = TarcerRecurse[expr]];t1=ta0 /. ints/.any_:>any;vars = Select[Variables[t1], Length[#1] > 0 &];If[!(FreeQ[vars,TFI|TVI|TJI|TKI|TBI|TAI]),Print["Returning the list of remaining \
integrals."];Throw[Select[vars,!(FreeQ[#1,TFI|TVI|TJI|TKI|TBI|TAI])&]]];t1 = t1 /. d -> fe;polygafu[2, 1] = -2 Zeta[3];polygafu[2, 2] = 2 - 2 Zeta[3];polygafu[x_, y_] := polygafu[x, y] = 
                  FunctionExpand[PolyGamma[x, y]];t2 = 
                Normal[t1 + O[e]^(order + 1)] /. 
                  PolyGamma -> polygafu;t3=Collect[t2, e, Expand[#1] /. z2rule &]//.{a_ b_:>a b}/.c_:>c;prec=Union[Cases[t3,_,-1]];If[Length[prec] === 1,If[Head[t3] === Plus,mass = 
                          Select[
                            Variables[
                              prec[[1, 
                                1]]], 
                            FreeQ[#1, SEpsilon] &];If[Length[mass] === 1, 
                          If[FreeQ[First[t3], First[mass]], massfactor = 1, 
                            massfactor = 
                              First[mass]^Exponent[First[t3], First[mass]]], 
                          massfactor = 1];delpfactor = 1;delp=Variables[First[t3]/massfactor/.{__:>1,ToExpression["HighEnergyPhysics`\
general`Zeta2`Zeta2"] -> Zeta[2]}];If[Length[delp] === 1, delp = First[delp]; 
                          If[Length[t3] === 
                              Length[
                                Select[
                                  t3, ! (FreeQ[#1, 
                                        delp]) &]], 
                            delpfactor = (delp^Exponent)[First[t3], delp]]];\(t3 = \((\(massfactor\ delpfactor\ First[prec] /. 
                                  Prefactor1 -> 
                                    Identity\) /. {\((m_\^2)\)\^n_\ m_\^j_?EvenQ :> \((m\^2)\)\^\(n + j\/2\)})\)\ Collect[\((#1\/\(First[prec]\ massfactor\ delpfactor\) &)\) /@ t3, e, 
                              Expand]\)]];t3 /. Prefactor1 -> Identity]]

Construct T's

remember

The function remember is not used by default.

tfch = {TFR :> Hold[TFR], TVR :> Hold[TVR], TJR :> Hold[TJR], 
        TBR :> Hold[TBR], TAR :> Hold[TAR], 
        TComment :> Hold[TComment]};

rtfch = Map[Reverse, tfch];

SetAttributes[condrest, HoldRest];

Clear[remember];

remember[zi_HoldForm] := 
      Block[{z = zi /. tfch}, 
        If[! (FreeQ[z, 
                Condition]), Hold[
                      HoldForm][Hold[SetDelayed][
                      z[[1, 1]], 
                      ReplacePart[
                        z[[1, 2]], 
                        Prepend[
                          Extract[z, {1, 2, 1}, 
                            Hold], z[[1, 
                                    1]] /. 
                                  PatternTest -> pt /. 
                                pt[a_, b_] :> a /. 
                              Pattern -> pat /. 
                            pat[a_, b_] :> a], 1]]] /. 
                  Hold[HoldForm] -> HoldForm /. rtfch /. 
              Hold[SetDelayed] -> SetDelayed /. 
            Hold -> 
              Set, Hold[HoldForm][Hold[SetDelayed][
                      z[[1, 1]], 
                      Prepend[
                        Extract[z, {1, 2}, 
                          Hold], z[[1, 
                                  1]] /. 
                                PatternTest -> pt /. 
                              pt[a_, b_] :> a /. 
                            Pattern -> pat /. 
                          pat[a_, b_] :> a]]] /. 
                  Hold[HoldForm] -> HoldForm /. rtfch /. 
              Hold[SetDelayed] -> SetDelayed /. 
            Hold -> Set]];

$TarasovTdeltaplimit

\(aa2\  = \ 
      Array[f\_# &, \ 
        10000 + \(($TarasovTdeltaplimit + 1)\)\ 1000]; \)

bb12 = Select[aa2, Head[#] =!= Subscript &];

Clear[aa2];

\(Print["there are ", 
      Length[bb12], " recursion equations for $RankLimit = ", $RankLimit]\)

If[Global`$Remember  ===  True, Global`RCHECK = True;  
      nb  = Map[remember, bb12], nb = bb12];

Clear[bb12];

Unset[$Post]

ReleaseHold[nb];

$Comment = False;

End[];

EndPackage[];

If[! ValueQ[HighEnergyPhysics`FeynCalc`$FeynCalcDirectory], 
        BeginPackage["HighEnergyPhysics`general`Zeta2`"]];

\(Zeta2::usage = 
"Zeta2 denotes Zeta[2]."; \)

Begin["`Private`"];

Zeta2  /:  N[Zeta2]  =  N[Zeta[2]]; 
Zeta2 /: MakeBoxes[Zeta2, fmt_] := 
      InterpretationBox @@ {  RowBox[{"ζ", "(", 2, ")"}], 
          Zeta2, Editable -> False};

End[];

EndPackage[];

$BasisIntegrals

This a (incomplete) set of basis integrals. You can add your favorite integrals here.

$BasisIntegrals = 
  {TAI[d_, 0, {{1, M_}}] :> 
    -I*E^(1/2*(4 - d)*EulerGamma)*M^2*
     Gamma[-1 + (4 - d)/2]*
     Prefactor1[(M^2)^(1/2*(-4 + d))*
       SEpsilon[d]], 
       
       TBI[d_, pp_, {{1, 0},{1, 0}}] :> 
   (I*E^(((4 - d)*EulerGamma)/2 - 
         I/2*(-4 + d)*Pi)*Gamma[2 - d/2]*
      Gamma[-1 + d/2]^2*
      Prefactor1[(pp)^((-4 + d)/2)*
        SEpsilon[d]])/Gamma[-2 + d],
        
        TJI[d_, pp_, {{1,0},{1,0},{1,0}}] :>-((pp*Gamma[3 - d]*Gamma[-1 + d/2]^3*
    E^((4 - d)*EulerGamma)*  Prefactor1[
        (pp)^(-4 + d)*SEpsilon[d]^2])/
    (E^(I*(-4 + d)*Pi)*Gamma[-3 + (3*d)/2])
),
    
          TJI[d_, M_^2, {{1,M_},{1,0},{1,0}}] :>(M^2*Gamma[3 - d]*Gamma[2 - d/2]*
    Gamma[-1 + d/2]^2*Gamma[-5 + 2*d]*E^((4 - d)*EulerGamma)*
    Prefactor1[
      (M^2)^(-4 + d)*SEpsilon[d]^2])/
  (Gamma[-2 + d]*Gamma[-3 + (3*d)/2]),
  
  TJI[d_Symbol, (M_)^2, 
  {{1, M_}, {1, M_}, {1, M_}}]:>M^2*Prefactor1[
    (M^2)^(-4 + d)*SEpsilon[d]^2]*
  (-6/(-4 + d)^2 + 17/(2*(-4 + d)) + 
    (-59 - 2*Pi^2)/8 + 
    (-4 + d)^3*TJI111e + 
    ((-4 + d)^2*(16755 - 4750*Pi^2 - 
         14*Pi^4 + 3840*Pi^2*Log[2] - 
         12080*Zeta[3]))/1920 + 
    ((-4 + d)*(195 + 98*Pi^2 - 
         48*Zeta[3]))/96),
         
         TJI[d_, 0, {{1, M_}, {1, M_}, {1, 0}}]:>(M^2*Gamma[3 - d]*Gamma[2 - d/2]^2*
    Gamma[-1 + d/2]*E^((4 - d)*EulerGamma)*
    Prefactor1[
      (M^2)^(-4 + d)*SEpsilon[d]^2])/
  (Gamma[4 - d]*Gamma[d/2]),
  
  TJI[d_, 0, {{1,M_},{1,0},{1,0}}] :>(M^2*Gamma[3 - d]*Gamma[2 - d/2]*
    Gamma[-1 + d/2]^2*E^((4 - d)*EulerGamma)*
    Prefactor1[
      (M^2)^(-4 + d)*SEpsilon[d]^2])/
  Gamma[d/2]
        }

Save tarcer*.mx file

Share[];

DumpSave[StringJoin["tarcer", StringJoin @@ 
     ToString /@ $RankLimit, ".mx"], 
   {$ContextPath, HighEnergyPhysics`Tarcer`Cayley, 
    HighEnergyPhysics`Tarcer`CayleyD, 
    HighEnergyPhysics`Tarcer`Cayleyu, 
    HighEnergyPhysics`Tarcer`Prefactor1, 
    HighEnergyPhysics`Tarcer`TarcerRecurse, 
    HighEnergyPhysics`Tarcer`TFIRecurse, 
    HighEnergyPhysics`Tarcer`TFISimplify, 
    HighEnergyPhysics`Tarcer`SEpsilon, 
    HighEnergyPhysics`Tarcer`TarcerExpand, 
    HighEnergyPhysics`Tarcer`TarcerReduce, 
    HighEnergyPhysics`Tarcer`TAI, 
    HighEnergyPhysics`Tarcer`TAR, 
    HighEnergyPhysics`Tarcer`TBI, 
    HighEnergyPhysics`Tarcer`TBR, 
    HighEnergyPhysics`Tarcer`TFI, 
    HighEnergyPhysics`Tarcer`TFR, 
    HighEnergyPhysics`Tarcer`TJI, 
    HighEnergyPhysics`Tarcer`TJR, 
    HighEnergyPhysics`Tarcer`TKI, 
    HighEnergyPhysics`Tarcer`TVI, 
    HighEnergyPhysics`Tarcer`TVR, 
    HighEnergyPhysics`Tarcer`$BasisIntegrals, 
    HighEnergyPhysics`Tarcer`$Comment, 
    HighEnergyPhysics`Tarcer`$CommentNotebook, 
    HighEnergyPhysics`Tarcer`$RankLimit, 
    HighEnergyPhysics`Tarcer`$TTable, 
    HighEnergyPhysics`Tarcer`Private`applytlrules, 
    HighEnergyPhysics`Tarcer`Private`applytlrules2,
    HighEnergyPhysics`Tarcer`Private`ExpandMaybe, 
    HighEnergyPhysics`Tarcer`Private`FactorC, 
    HighEnergyPhysics`Tarcer`Private`FEPrint, 
    HighEnergyPhysics`Tarcer`Private`fN, 
    HighEnergyPhysics`Tarcer`Private`mbt, 
    HighEnergyPhysics`Tarcer`Private`PNQ, 
    HighEnergyPhysics`Tarcer`Private`PQ, 
    HighEnergyPhysics`Tarcer`Private`ta, 
    HighEnergyPhysics`Tarcer`Private`TA, 
    HighEnergyPhysics`Tarcer`Private`TAIC, 
    HighEnergyPhysics`Tarcer`Private`TBIC, 
    HighEnergyPhysics`Tarcer`Private`TComment, 
    HighEnergyPhysics`Tarcer`Private`TLRComment, 
    HighEnergyPhysics`Tarcer`Private`TFIC, 
    HighEnergyPhysics`Tarcer`Private`TJIC, 
    HighEnergyPhysics`Tarcer`Private`TLR, 
    HighEnergyPhysics`Tarcer`Private`tlrule, 
    HighEnergyPhysics`Tarcer`Private`redblue, 
    HighEnergyPhysics`Tarcer`Private`TLRules, 
    HighEnergyPhysics`Tarcer`Private`TLRules2, 
    HighEnergyPhysics`Tarcer`Private`TVIC, 
    HighEnergyPhysics`Tarcer`Private`$TLRComment, 
    HighEnergyPhysics`general`Zeta2`Zeta2}];

$HistoryLength = ∞;

This gives the absolute time (in minutes) to run this notebook.

Round[AbsoluteTime[] - HighEnergyPhysics`Tarcer`Private`atime]/
        60.  min

Display the size of the generated file:

filename  =  "tarcer" <> (Apply[StringJoin, 
            ToString /@ $RankLimit]) <> ".mx"

FileByteCount[filename] Byte

Directory[]

MaxMemoryUsed[]

MemoryInUse[]

TimeUsed[]