<<PolylogTools`

rep[tensor_, power_] := Sequence@@Table[tensor, {power}]
symbLog[argument_, power_] := power! CiTi[rep[argument, power]]

symbmLog[lambda_, argument_, power_] := lambda^power symbLog[argument, power]/power!

symbIab[{a_,b_}, {x_,y_}] := Sum[(-1)^(t1 + b - 1) Tens[Log[1 - 1/y], Shuffle[Tens[rep[Log[y], b-1]], Tens[Log[1-y/x], Shuffle[Tens[rep[Log[x], t1]], Tens[rep[Log[y], a-1 - t1]]]]]], {t1, 0, a - 1}] + Sum[(-1)^(t1 + b) Binomial[a - 1 - t1 + b - 1 - t3, a - 1 - t1]Tens[Log[1 - 1/x], rep[x, t3], Log[1 - x/y], Shuffle[Tens[rep[Log[y], a - 1 - t1  +  b - 1 - t3]], Tens[rep[Log[x], t1]]]], {t1, 0, a - 1}, {t3, 0, b - 1}] +
Sum[(-1)^(t1 + b + 1) Binomial[a - 1 - t1 + b - 1, a - 1 - t1]Tens[Log[1 - 1/x], Shuffle[Tens[rep[Log[x], t1]], Tens[Log[1 - 1/y], rep[Log[y], a - 1 - t1 + b - 1]]]], {t1, 0, a - 1}]/.{Tens->CiTi,Log[j_] :>j}//ReduceTensorProduct

symbLi[n_, arg_] := -CiTi[1-arg, rep[arg, n-1]]

tsymbmLog[lambda_, arg_, power_] := lambda^power Tens[rep[Log[arg], power]]

tsymbLi[n_, arg_] := -Tens[Log[1-arg], rep[Log[arg], n-1]]

(* ------------ *)

coeffs[n_,i_] := Binomial[n+i,n]

rhsterms[a_,b_] := (-1)^b Sum[coeffs[b-1, i]Shuffle[tsymbmLog[-1, x, a-i], tsymbmLog[1, y, b+i]], {i, 0, a}] + 
(-1)^b Sum[coeffs[b-1,i] Shuffle[tsymbmLog[-1, x, a-i], tsymbLi[b+i, y]], {i,0,a}] +
-1*(-1)^a Sum[coeffs[a-1,i] Shuffle[tsymbmLog[-1, x, b-i], tsymbLi[a+i, x/y]], {i, 0, b}] +
(-1)^(a+b)Shuffle[ tsymbLi[a,x/y] , tsymbmLog[1,y,b]] +
(-1)^(a+b) tsymbLi[a+b, x]/.{Tens->CiTi,Log[j_] :>j}//ReduceTensorProduct

lhsterms[a_,b_] := symbIab[{a,b}, {x,y}] - (-1)^(a+b) symbIab[{a,b}, {1/x,1/y}]

rhsterms[2,3]

lhsterms[2,3]

lhsterms[2,3] - rhsterms[2,3]

lhsterms[5,4] - rhsterms[5,4]

lhsterms[12,2]

rhsterms[12,2]

%145 - %146

lhsterms[5,7] - rhsterms[5,7]

(* -------------------- *)

symbIab2[{a_,b_}, {x_,y_}] := Sum[(-1)^(t1 + b - 1) Tens[Log[1 - 1/y], Shuffle[Tens[rep[Log[y], b-1]], Tens[Log[1-y/x], Shuffle[Tens[rep[Log[x], t1]], Tens[rep[Log[y], a-1 - t1]]]]]], {t1, 0, a - 1}] + Sum[(-1)^(t1 + b) Binomial[a - 1 - t1 + b - 1 - t3, a - 1 - t1]Tens[Log[1 - 1/x], rep[x, t3], Log[1 - x/y], Shuffle[Tens[rep[Log[y], a - 1 - t1  +  b - 1 - t3]], Tens[rep[Log[x], t1]]]], {t1, 0, a - 1}, {t3, 0, b - 1}] +
0Sum[(-1)^(t1 + b + 1) Binomial[a - 1 - t1 + b - 1, a - 1 - t1]Tens[Log[1 - 1/x], Shuffle[Tens[rep[Log[x], t1]], Tens[Log[1 - 1/y], rep[Log[y], a - 1 - t1 + b - 1]]]], {t1, 0, a - 1}]/.{Tens->CiTi,Log[j_] :>j}//ReduceTensorProduct

rhsterms2[a_,b_] := 0(-1)^b Sum[coeffs[b-1, i]Shuffle[tsymbmLog[-1, x, a-i], tsymbmLog[1, y, b+i]], {i, 0, a}] + 
0(-1)^b Sum[coeffs[b-1,i] Shuffle[tsymbmLog[-1, x, a-i], tsymbLi[b+i, y]], {i,0,a}] +
-1*(-1)^a Sum[coeffs[a-1,i] Shuffle[tsymbmLog[-1, x, b-i], tsymbLi[a+i, x/y]], {i, 0, b}] +
(-1)^(a+b)Shuffle[ tsymbLi[a,x/y] , tsymbmLog[1,y,b]] +
0(-1)^(a+b) tsymbLi[a+b, x]/.{Tens->CiTi,Log[j_] :>j}//ReduceTensorProduct

lhsterms2[a_,b_] := symbIab2[{a,b}, {x,y}] - (-1)^(a+b) symbIab2[{a,b}, {1/x,1/y}]

lhsterms2[4,2]

rhsterms2[4,2]

Intersection[%163, %164]