Discovering and Proving Infinite Pochhammer Sum Identities

This notebook (which has been tested using the Mathematica versions 11.3 and 12.0) accompanies the article:
 J. Ablinger. Discovering and Proving Infinite Pochhammer Sum Identities. [arXiv:1902.11001 [math.CO]].
 To start we have to load the package HarmonicSums.m, which can be downloaded from here.

<<HarmonicSums.m

To see a full list of available commands in HarmonicSums type

HarmonicSumsFunctionList[]

Several precomputed tables will be used during the computations:
RelTabSinfH1,
RelTabCSinf,
RelTabCycloH4ToSinf.
These tables should be placed in the directory, where you saved the packages HarmonicSums and will loaded in automatically. You can get these tables here, together with the table RelTabCycloH6tow4.m, which is will be used at some places in this notebook. We now read in the RelTabCycloH6tow4.m:

hrel=<<RelTabCycloH6tow4.m;

Now we are ready to give examples for the 3 main commands described in the article:

Examples

GeneratingFunciton

The documentation of ComputeGeneratingFunction can be displayed by typing

?ComputeGeneratingFunction

Let us state some examples now:

ComputeGeneratingFunction[((Pochhammer[-(1/2),n]S[1,n])/((n+3)^2(n-1)!)),x,{n,1,∞}]
%/.x->1//SpecialGLToH//Expand//GLToH
%//ReduceConstants[#,ToKnownConstants->True]&//Expand

ComputeGeneratingFunction[((Pochhammer[(1/3),n]S[1,1,1,n])/((n+1)!)),x,{n,1,∞}]
%/.x->1//SpecialGLToH//Expand//GLToH
%//ReduceConstants[#,ToKnownConstants->True]&//Expand
ToCyclotomic[%]/.hrel//Expand

ComputeGeneratingFunction[((Pochhammer[(1/3),n]S[2,n])/((n+1)!)),x,{n,1,∞}]
%/.x->1//SpecialGLToH//Expand//GLToH
%//ReduceConstants[#,ToKnownConstants->True]&//Expand
ToCyclotomic[%]/.hrel//Expand

ComputeGeneratingFunction[((Pochhammer[(1/4),n]S[2,n])/((n+1)!)),x,{n,1,∞}]
%/.x->1//SpecialGLToH//Expand//GLToH
%//ReduceConstants[#,ToKnownConstants->True]&//Expand

ComputeGeneratingFunction[((Pochhammer[(1/2),n]CS[{{2,1,1}},n])/((2n+1)^2n!)),x,{n,1,∞}]
%/.x->1//SpecialGLToH//Expand//GLToH
%//ReduceConstants[#,ToKnownConstants->True]&//Expand

ComputeGeneratingFunction[((Pochhammer[(1/2),n]CS[{{2,1,1},{2,1,1}},n])/((2n+1)^2n!)),x,{n,1,∞}]
%/.x->1//SpecialGLToH//Expand//GLToH
%//ReduceConstants[#,ToKnownConstants->True]&//Expand

ComputeGeneratingFunction[((Pochhammer[3,n]S[3,n])/(n^2n!)),x,{n,1,∞}]
%/.x->1/2//SpecialGLToH//Expand//GLToH
%//ReduceConstants[#,ToKnownConstants->True]&//Expand

PochhammerSumToGL

The documentation of PochhammerSumToGL can be displayed by typing

?PochhammerSumToGL

Let us state some examples now:

PochhammerSumToGL[((Pochhammer[-(1/2),n]S[1,n])/((n+3)^2(n-1)!)),x,{n,1,∞}]
%/.x->1//SpecialGLToH//Expand//GLToH//ToNotCyclotomic
%//ReduceConstants//Expand

PochhammerSumToGL[Pochhammer[1/3,n]/(n+1)! S[1,1,n],x,{n,1,∞}]
%//GLToH//RemoveLeading1//CollectTogether
%/.x->1//SpecialGLToH//GLToH//ReduceConstants//Expand
ToCyclotomic[%]/.hrel//Expand

PochhammerSumToGL[Pochhammer[1/2,n]/(n+1)! S[1,1,1,1,1,n],x,{n,1,∞}]
%//GLToH//RemoveLeading1//CollectTogether
%/.x->1//SpecialGLToH//GLToH//ReduceConstants//Expand

PochhammerSumToGL[Pochhammer[1/2,n]/n/(n+1)! S[4,n],x,{n,1,∞}]
%//GLToH//RemoveLeading1//CollectTogether
%/.x->1//SpecialGLToH//GLToH//ReduceConstants//Expand

PochhammerSumToGL[Pochhammer[-1/2,n]CS[{{3,1,1}},n]/n/(n-1)! ,x,{n,1,∞}]
%//GLToH//RemoveLeading1//CollectTogether
%/.x->1//SpecialGLToH//GLToH//ReduceConstants//ToHarmonicSumsIntegrate//ReleaseHold//FullSimplify

PochhammerSumToH

The documentation of  PochhammerSumToH can be displayed by typing

?PochhammerSumToH

Let us state some examples now:

PochhammerSumToH[((Pochhammer[-1/2,n]S[1,n])/((n+3)^2(n-1)!)),x,{n,1,∞}]
%/.x->1/.H[-1,0]->0/.H[1,0]->0/. H[-1,0,0]->0/.H[1,0,0]->0/.H[1,-1,0]->0

PochhammerSumToH[((Pochhammer[-1/2,n]S[1,n])/((n+3)^2(n-1)!)),x,{n,1,∞}]
%/.x->1-Eta
%/.Eta->0//ReduceConstants//Expand

PochhammerSumToH[Pochhammer[1/5,n]/(n+1)! S[1,n],x,{n,1,∞}]
%//GLToH//RemoveLeading1//CollectTogether
%/.x->1//GLRemoveTrailing0//GLToH//Expand//ReduceConstants//Expand

PochhammerSumToH[Pochhammer[1/2,n]/(n+2)! S[2,2,n],x,{n,1,∞}]
%//GLToH//RemoveLeading1//CollectTogether
%/.x->1//GLRemoveTrailing0//GLToH//Expand//ReduceConstants//Expand

PochhammerSumToH[Pochhammer[1/2,n]/n/(n+2)! S[2,2,n],x,{n,1,∞}]
%//GLToH//RemoveLeading1//CollectTogether
%/.x->1//GLRemoveTrailing0//GLToH//Expand//ReduceConstants//Expand

PochhammerSumToH[Pochhammer[1/2,n]/(n+1)! S[5,n],x,{n,1,∞}]
%//GLToH//RemoveLeading1//CollectTogether
%/.x->1//GLRemoveTrailing0//GLToH//Expand//ReduceConstants//Expand

Constants

Note that in the article
 J. Ablinger. Discovering and Proving Infinite Pochhammer Sum Identities. [arXiv:1902.11001 [math.CO]].
 we use for the subclass of multiple zeta values and Euler sums a slightly different, but equivalent, set constants compared to
 J. Blümlein,  D.J. Broadhurst, and J.A.M. Vermaseren, The Multiple Zeta Value Data Mine, Comput. Phys. Commun. 181 (2010) 582-625.
 In the following we state replacement rules, how the constants for the article can be rewritten in terms of the constants used in the multiple zeta data mine, and vice versa.

({{(Subscript[l, 2])->ln2}, {(Subscript[ζ, 3])->((4 e3)/3)}, {(Subscript[ζ, 5])->((16 e5)/15)}, {(Subscript[ζ, 7])->((64 e7)/63)}, {(Subscript[ζ, 9])->((256 e9)/255)}, {(Subscript[ζ, 11])->((1024 e11)/1023)}, {(Subscript[p, 4])->(h31/2)-((7 e3 ln2)/6)-((ln2^4)/24)+(((ln2^2) z2)/4)+((3 (z2^2))/8)}, {(Subscript[p, 5])->((11 e5)/20)+(h311/2)-((h31 ln2)/2)+((7 e3 (ln2^2))/12)+((ln2^5)/120)+((e3 z2)/3)-(((ln2^3) z2)/12)-((3 ln2 (z2^2))/8)}, {(Subscript[p, 6])->((2 (e3^2))/9)+(h3111/2)-(h51/4)-((11 e5 ln2)/20)-((h311 ln2)/2)+((h31 (ln2^2))/4)-((7 e3 (ln2^3))/36)-((ln2^6)/720)-((e3 ln2 z2)/3)+(((ln2^4) z2)/48)+((3 (ln2^2) (z2^2))/16)+((53 (z2^3))/280)}, {(Subscript[p, 7])->((53 e7)/42)+(h31111/2)-(h511/4)-((2 (e3^2) ln2)/9)-((h3111 ln2)/2)+((h51 ln2)/4)+((11 e5 (ln2^2))/40)+((h311 (ln2^2))/4)-((h31 (ln2^3))/12)+((7 e3 (ln2^4))/144)+((ln2^7)/5040)+(1/6) e3 (ln2^2) z2-(((ln2^5) z2)/240)-((e3 (z2^2))/10)-(((ln2^3) (z2^2))/16)-((53 ln2 (z2^3))/280)}, {(Subscript[p, 8])->(h311111/2)-(h5111/4)+(h71/2)-((53 e7 ln2)/42)-((h31111 ln2)/2)+((h511 ln2)/4)+(((e3^2) (ln2^2))/9)+((h3111 (ln2^2))/4)-((h51 (ln2^2))/8)-((11 e5 (ln2^3))/120)-((h311 (ln2^3))/12)+((h31 (ln2^4))/48)-((7 e3 (ln2^5))/720)-((ln2^8)/40320)-((2 (e3^2) z2)/9)-(1/18) e3 (ln2^3) z2+(((ln2^6) z2)/1440)+(1/10) e3 ln2 (z2^2)+(((ln2^4) (z2^2))/64)+((53 (ln2^2) (z2^3))/560)+((397 (z2^4))/2240)}, {(Subscript[p, 9])->-((8 (e3^3))/81)-((1321 e9)/1224)+(h3111111/2)-(h51111/4)+(h711/2)-((h311111 ln2)/2)+((h5111 ln2)/4)-((h71 ln2)/2)+((53 e7 (ln2^2))/84)+((h31111 (ln2^2))/4)-((h511 (ln2^2))/8)-(((e3^2) (ln2^3))/27)-((h3111 (ln2^3))/12)+((h51 (ln2^3))/24)+((11 e5 (ln2^4))/480)+((h311 (ln2^4))/48)-((h31 (ln2^5))/240)+((7 e3 (ln2^6))/4320)+((ln2^9)/362880)+((8 e7 z2)/21)+(2/9) (e3^2) ln2 z2+(1/72) e3 (ln2^4) z2-(((ln2^7) z2)/10080)+((7 e5 (z2^2))/25)-(1/20) e3 (ln2^2) (z2^2)-(((ln2^5) (z2^2))/320)+((83 e3 (z2^3))/420)-((53 (ln2^3) (z2^3))/1680)-((397 ln2 (z2^4))/2240)}, {(Subscript[s, 1])->((4 (e3^2))/3)-h51+((31 e5 ln2)/15)-((23 (z2^3))/70)}, {(Subscript[s, 2])->-((287 e7)/102)+((7 h331)/17)-((25 h511)/17)+((11 (e3^2) ln2)/6)-h51 ln2+((31 e5 (ln2^2))/15)+((1319 e5 z2)/510)-((103 e3 (z2^2))/340)-((141 ln2 (z2^3))/280)}, {(Subscript[s, 3])->((247 e7)/126)-h511-((11 e5 z2)/10)-((7 e3 (z2^2))/15)}, {(Subscript[s, 4])->-((6400 e3 e5)/513)-((320 h53)/513)+((1600 h71)/171)+((48742 (z2^4))/29925)}, {(Subscript[s, 5])->((12 e3 e5)/5)-h71+((127 e7 ln2)/63)-((951 (z2^4))/2800)}, {(Subscript[s, 6])->((135041 e3 e5)/174420)-((7 h3311)/17)+((25 h5111)/17)+((17075 h53)/34884)-((4841 h71)/684)-((491 e7 ln2)/102)+((7 h331 ln2)/17)-((25 h511 ln2)/17)+((11 (e3^2) (ln2^2))/6)-h51 (ln2^2)+((62 e5 (ln2^3))/45)+((707 (e3^2) z2)/204)-((h51 z2)/2)+((621 e5 ln2 z2)/170)+((79 e3 ln2 (z2^2))/2040)-((141 (ln2^2) (z2^3))/280)-((45742853 (z2^4))/65116800)}, {(Subscript[s, 7])->-((40543 e3 e5)/5130)+((4 e3 h311)/3)-h5111+((335 h53)/1026)+((1061 h71)/342)-((232 e7 ln2)/21)-((4 e3 h31 ln2)/3)+h511 ln2+((10 (e3^2) (ln2^2))/9)-((h51 (ln2^2))/2)+((31 e5 (ln2^3))/90)+((34 (e3^2) z2)/9)-((5 h51 z2)/2)+((187 e5 ln2 z2)/30)-((h31 (z2^2))/10)+(7/10) e3 ln2 (z2^2)-((51 (ln2^2) (z2^3))/140)+((728653 (z2^4))/1915200)}})

({{e3->((3 z3)/4)}, {h31->2 li4half+((ln2^4)/12)-(((ln2^2) z2)/2)-((3 (z2^2))/4)+((7 ln2 z3)/4)}, {e5->((15 z5)/16)}, {h311->2 li5half+2 li4half ln2+((ln2^5)/15)-(((ln2^3) z2)/3)+((7 (ln2^2) z3)/8)-((z2 z3)/2)-((33 z5)/32)}, {h51->-((23 (z2^3))/70)+((3 (z3^2))/4)+((31 ln2 z5)/16)-(Subscript[s, 1])}, {h3111->2 li6half+2 li5half ln2+li4half (ln2^2)+((ln2^6)/36)-(((ln2^4) z2)/8)-((19 (z2^3))/35)+((7 (ln2^3) z3)/24)+((z3^2)/8)+((31 ln2 z5)/32)-((Subscript[s, 1])/2)}, {e7->((63 z7)/64)}, {h511->-((7 (z2^2) z3)/20)-((33 z2 z5)/32)+((247 z7)/128)-(Subscript[s, 3])}, {h331->((17 ln2 (z2^3))/40)-((391 (z2^2) z3)/560)-((153 ln2 (z3^2))/224)-((67 z2 z5)/7)+((6101 z7)/448)-((17 ln2 (Subscript[s, 1]))/7)+((17 (Subscript[s, 2]))/7)-((25 (Subscript[s, 3]))/7)}, {h31111->2 li7half+2 li6half ln2+li5half (ln2^2)+((li4half (ln2^3))/3)+((ln2^7)/126)-(((ln2^5) z2)/30)+((7 (ln2^4) z3)/96)-(((z2^2) z3)/40)-((33 z2 z5)/64)-((389 z7)/256)-((Subscript[s, 3])/2)}, {h71->-((951 (z2^4))/2800)+((27 z3 z5)/16)+((127 ln2 z7)/64)-(Subscript[s, 5])}, {h53->-((69537 (z2^4))/28000)+((45 z3 z5)/4)+((1905 ln2 z7)/64)-((513 (Subscript[s, 4]))/320)-15 (Subscript[s, 5])}, {h5111->-((li4half (z2^2))/5)-(((ln2^4) (z2^2))/120)-((3 (ln2^2) (z2^3))/20)-((3291 (z2^4))/5600)+2 li5half z3-(((ln2^5) z3)/60)+(1/6) (ln2^3) z2 z3+(3/4) ln2 (z2^2) z3-((5 (ln2^2) (z3^2))/8)-((z2 (z3^2))/4)-((31 (ln2^3) z5)/48)-((ln2 z2 z5)/32)+((297 z3 z5)/128)+((887 ln2 z7)/128)+(((ln2^2) (Subscript[s, 1]))/2)+((5 z2 (Subscript[s, 1]))/2)-ln2 (Subscript[s, 3])-((67 (Subscript[s, 4]))/128)-8 (Subscript[s, 5])-(Subscript[s, 7])}, {h3311->-((5 li4half (z2^2))/7)-((5 (ln2^4) (z2^2))/168)-((15 (ln2^2) (z2^3))/28)-((30589 (z2^4))/58800)+((50 li5half z3)/7)-((5 (ln2^5) z3)/84)+(25/42) (ln2^3) z2 z3+((3697 ln2 (z2^2) z3)/1120)-((125 (ln2^2) (z3^2))/56)+((1313 z2 (z3^2))/448)-((31 (ln2^3) z5)/8)-((ln2 z2 z5)/28)-((10793 z3 z5)/1792)+((9517 ln2 z7)/448)+((25 (ln2^2) (Subscript[s, 1]))/14)+((71 z2 (Subscript[s, 1]))/7)+((17 ln2 (Subscript[s, 2]))/7)-((25 ln2 (Subscript[s, 3]))/7)-((6765 (Subscript[s, 4]))/1792)-((409 (Subscript[s, 5]))/14)-((17 (Subscript[s, 6]))/7)-((25 (Subscript[s, 7]))/7)}, {h311111->2 li8half+2 li7half ln2+li6half (ln2^2)+((li5half (ln2^3))/3)+((li4half (ln2^4))/12)+((ln2^8)/576)-(((ln2^6) z2)/144)-((li4half (z2^2))/10)-(((ln2^4) (z2^2))/240)-((3 (ln2^2) (z2^3))/40)-((3457 (z2^4))/11200)+li5half z3+(((ln2^5) z3)/160)+(1/12) (ln2^3) z2 z3+(3/8) ln2 (z2^2) z3-((5 (ln2^2) (z3^2))/16)+((z2 (z3^2))/8)-((31 (ln2^3) z5)/96)-((ln2 z2 z5)/64)-((135 z3 z5)/256)+((379 ln2 z7)/256)+(((ln2^2) (Subscript[s, 1]))/4)+((5 z2 (Subscript[s, 1]))/4)-((ln2 (Subscript[s, 3]))/2)-((67 (Subscript[s, 4]))/256)-3 (Subscript[s, 5])-((Subscript[s, 7])/2)}, {e9->((255 z9)/256)}, {e11->((1023 z11)/1024)}})