page 272-Check on Generalized Laguerre-Forsyth Identity

The version number of Mathematica used to evaluate this notebook:

$VersionNumber

with copy and paste from page 272   (for n = 3 and p = 5)

p  =  5;

c[i2_,i1_,i3_,k_] := c[i1, i2, i3, k]  /;  i2  >  i1

c[i1_,i3_,i2_,k_] := c[i1, i2, i3, k]  /;  i3  >  i2

exprD[i1_,j1_,i2_,j2_,i3_,j3_,k_,r_] := 
        Binomial[k, r]*Product[2*i1 + 2*i2 + 2*i3 - j1 - j2 - j3 + k - t,{t, 1, r}]*
        (j1!)*Binomial[i1 - 1, j1]*Binomial[m - i1 + j1, j1]*
        (j2!)*Binomial[i2 - 1, j2]*Binomial[m - i2 + j2, j2]*
        (j3!)*Binomial[i3 - 1, j3]*Binomial[m - i3 + j3, j3]

d[i1_,i2_,i3_,k_] := (F)^(i1 + i2 + i3 + k)*Sum[ (T)^(j1 + j2 + j3 + r)*
        exprD[i1, j1, i2, j2, i3, j3, k, r]*c[i1 - j1, i2 - j2, i3 - j3, k - r],
        {j1, 0, i1},{j2, 0, i2},{j3, 0, i3},{r, 0, k}]

A[i1_,j1_,i2_,j2_,i3_,j3_]:=(-1)^(j1 + j2 + j3)*Binomial[m - i1 + j1, j1]*Binomial[m - i2 + j2, j2]*Binomial[m - i3 + j3, j3]*Product[i1 - k,{k, 1, j1}]*Product[i2 - k,{k, 1, j2}]*Product[i3 - k,{k, 1, j3}]/Product[2  i1 + 2  i2 + 2  i3 - 1 - k,{k, 1, j1 + j2 + j3}]/;2  i1 + 2  i2 + 2  i3 - 2 >= j1 + j2 + j3

Inv[i1_,i2_,i3_] := Sum[ \(A[i1, j1, i2, j2, i3, j3]*
\ \ \ \ \ \ \ \ c[i1 - j1, 
                i2 - j2, i3 - j3, j1 + j2 + j3]\),{j1, 0, i1},{j2, 0, i2},{j3, 0, i3}]

InvSharp[i1_,i2_,i3_] := Sum[ \(A[i1, j1, i2, j2, i3, j3]*
\ \ \ \ \ \ \ \ d[i1 - j1, 
                i2 - j2, i3 - j3, j1 + j2 + j3]\),{j1, 0, i1},{j2, 0, i2},{j3, 0, i3}]

difference[i1_,i2_,i3_] := FullSimplify[InvSharp[i1, i2, i3] 
                                     - (F)^(i1 + i2 + i3)*Inv[i1, i2, i3]]

Do[Print["difference[", i1, ", ", i2, ", ", 
              i3, "] = ", Expand[difference[i1, i2, i3]], ",     ", 
              Date[]],{i1, 0, p},{i2, i1, p},{i3, Max[i2, 2 - i1 - i2], p}]