<<HarmonicSums.m

Sun 1.40

We start by computing a closed form for the generating function of (((4/(k^5))+33 (Subscript[S, 5])(k))/((k^2) (Binomial[2 k, k]))):

s=ComputeGeneratingFunction[1/Binomial[2k,k]/k^2(33*S[5,k]+4/k^5),x,{k,1,∞}]

Next we send x->1:

r1=s/.x->1

Now we transform the iterated integrals to cyclotomic harmonic polylogarithms:

r2=r1//SpecialGLToH//GLToH//Expand

We can now use the command ComputeCycloH1Basis to compute the relations between the cyclotomic harmonic polylogarithms. Alternatively these relations can be downloaded form http://www.risc.jku.at/research/combinat/software/HarmonicSums.

(*Take care, this might take some time.*)
hrel=ComputeCycloH1Basis[7,3,ToBasis->{H[{0,0},{3,1},1],H[{0,0},{0,0},{1,0},1],H[{0,0},{3,1},{3,1},1],H[{0,0},{0,0},{0,0},{3,1},1],H[{0,0},{0,0},{3,1},{3,1},1],H[{0,0},{3,1},{3,1},{3,1},1],H[{0,0},{0,0},{0,0},{0,0},{1,0},1],H[{0,0},{0,0},{0,0},{0,0},{0,0},{0,0},{1,0},1]},NumberOfParallelKernels->20][[2]];

Applying this relations:

r3=r2/.hrel//Expand

Finally, rewriting this constants in terms of known constants leads to the result:

r3/.{H[{3,0},1]->(π/(3 (Sqrt[3]))),H[{0,0},{0,0},{1,0},1]->-z3,H[{0,0},{0,0},{0,0},{0,0},{1,0},1]->-z5,H[{0,0},{0,0},{0,0},{0,0},{0,0},{0,0},{1,0},1]->-z7}

Sun 1.41

We start by computing a closed form for the generating function of (((8/(k^3))+33 (Subscript[S, 3])(k))/((k^4) (Binomial[2 k, k]))):

s=ComputeGeneratingFunction[1/Binomial[2k,k]/k^4(33*S[3,k]+8/k^3),x,{k,1,∞}]

Next we send x->1:

r1=s/.x->1

Now we transform the iterated integrals to cyclotomic harmonic polylogarithms:

r2=r1//SpecialGLToH//GLToH//Expand

Applying the relations between cyclotomic harmonic polylogarithms:

r3=r2/.hrel//Expand

Finally, rewriting this constants in terms of known constants leads to the result:

r3/.{H[{3,0},1]->(π/(3 (Sqrt[3]))),H[{0,0},{0,0},{1,0},1]->-z3,H[{0,0},{0,0},{0,0},{0,0},{1,0},1]->-z5,H[{0,0},{0,0},{0,0},{0,0},{0,0},{0,0},{1,0},1]->-z7}

New Identities

We compute closed forms for the following expressions:

s={(Underoverscript[(∑^(.)), (k=1), ∞])(1/((k^7) Binomial[2 k,k])),(Underoverscript[(∑^(.)), (k=1), ∞])((S[1,k])/((k^6) Binomial[2 k,k])),(Underoverscript[(∑^(.)), (k=1), ∞])((S[2,k])/((k^5) Binomial[2 k,k])),(Underoverscript[(∑^(.)), (k=1), ∞])((S[3,k])/((k^4) Binomial[2 k,k])),(Underoverscript[(∑^(.)), (k=1), ∞])((S[4,k])/((k^3) Binomial[2 k,k])),(Underoverscript[(∑^(.)), (k=1), ∞])((S[5,k])/((k^2) Binomial[2 k,k])),(Underoverscript[(∑^(.)), (k=1), ∞])((S[6,k])/(k Binomial[2 k,k]))};

s1={ComputeGeneratingFunction[(1/((k^7) Binomial[2 k,k])),x,{k,1,∞}],ComputeGeneratingFunction[((S[1,k])/((k^6) Binomial[2 k,k])),x,{k,1,∞}],ComputeGeneratingFunction[((S[2,k])/((k^5) Binomial[2 k,k])),x,{k,1,∞}],ComputeGeneratingFunction[((S[3,k])/((k^4) Binomial[2 k,k])),x,{k,1,∞}],ComputeGeneratingFunction[((S[4,k])/((k^3) Binomial[2 k,k])),x,{k,1,∞}],ComputeGeneratingFunction[((S[5,k])/((k^2) Binomial[2 k,k])),x,{k,1,∞}],ComputeGeneratingFunction[((S[6,k])/(k Binomial[2 k,k])),x,{k,1,∞}]}

r1=s1/.x->1//CollectTogether

Now we transform the iterated integrals to cyclotomic harmonic polylogarithms:

r2=r1//SpecialGLToH//GLToH;

Applying the relations between cyclotomic harmonic polylogarithms:

r3=ToCyclotomic[r2]/.hrel//Expand;

constrel={H[{3,0},1]->(π/(3 (Sqrt[3]))),H[{3,1},1]->(1/18) (-(Sqrt[3]) π+9 Log[3]),H[{0,0},{3,1},1]->-1-c1+((7 (π^2))/54),H[{0,0},{0,0},{1,0},1]->-z3,H[{0,0},{3,1},{3,1},1]->1+(c2/9)+(π/(2 (Sqrt[3])))+((2 c1 π)/(3 (Sqrt[3])))-((14 (π^3))/(243 (Sqrt[3])))-((4 z3)/9)-((Log[3])/2),H[{0,0},{0,0},{0,0},{3,1},1]->-c3+((79 (π^4))/7290),H[{0,0},{0,0},{3,1},{3,1},1]->2 c1+(c1^2)+((3 c3)/2)+(c4/27)+(π/(6 (Sqrt[3])))-((4 (π^2))/27)-((c1 (π^2))/6)-((79 (π^4))/29160)-((26 π z3)/(81 (Sqrt[3])))+((Log[3])/2),H[{0,0},{0,0},{0,0},{0,0},{1,0},1]->-z5,H[{0,0},{0,0},{0,0},{0,0},{0,0},{0,0},{1,0},1]->-z7};
rep={c1->CS[{{3,1,2}},∞],c2->CS[{{1,0,2},{3,1,1}},∞],c3->CS[{{3,1,4}},∞]+1,c4->CS[{{1,0,3},{3,1,1}},∞]};

r4=r3/.constrel//Expand

Finally we try to combine this to get some nice identities:

r4.{a,b,c,d,e,f,g}/.a->-(c/3)/.b->0/.c->3/.d->0/.e->0/.f->0/.g->0//Expand
s.{a,b,c,d,e,f,g}/.a->-(c/3)/.b->0/.c->3/.d->0/.e->0/.f->0/.g->0//Expand

r4.{a,b,c,d,e,f,g}/.a->0/.b->0/.c->((8 d)/11)/.d->11/.e->0/.f->0/.g->0//Expand
s.{a,b,c,d,e,f,g}/.a->0/.b->0/.c->((8 d)/11)/.d->11/.e->0/.f->0/.g->0//Expand

r4.{a,b,c,d,e,f,g}/.a->0/.b->0/.c->0/.d->-1/.e->0/.f->2/.g->0//Expand
s.{a,b,c,d,e,f,g}/.a->0/.b->0/.c->0/.d->-1/.e->0/.f->2/.g->0//Expand