Zagier's Identity

Author

Eric W. Weisstein
January 10, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/ZagiersIdentity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ZagiersIdentity.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Identities

(∑_(n=0)^∞)((QPochhammer1[q])-(QPochhammer[q, q, n]))==(∑_r^∞)(((-1))^r) ((3 r-1) (q^( r (3 r-1)/2))+3 r (q^(r (3 r+1)/2)))+(((QPochhammer1[q]) ((QPolyGamma[1, q])+log(1-q)))/(log(q)))

Sum[QPochhammer[q]-QPochhammer[q,q,n],{n,0,200}]/.q->.1

Plot

With[{q=.1,inf=200},
QPochhammer[q]((QPolyGamma[1,q]+Log[1-q])/(Log[q]))+Sum[(((-1))^r)((3r-1)(q^(r(3r-1)/2))+3r (q^(r(3r+1)/2))),{r,inf}]
]

Sum[QPochhammer[q,q,∞]-QPochhammer[q,q,n],{n,0,∞}]

Series[Sum[QPochhammer[q,q,∞]-QPochhammer[q,q,n],{n,0,100}],{q,0,20}]

Sum[((q^k)/(1-(q^k))),{k,∞}]

Sum[(((-1))^r)((3r-1)(q^(r(3r-1)/2))+3r (q^(r(3r+1)/2))),{r,∞}]//Timing

Sum[(((-1))^r)(3r-1)(q^(r(3r-1)/2)),{r,∞}]//Timing

Sum[(((-1))^r)3 (q^((1/2) r (1+3 r))) r,{r,∞}]//Timing

FullSimplify[QPochhammer[q]((QPolyGamma[1,q]+Log[1-q])/(Log[q]))]

Series[QPochhammer[q]((QPolyGamma[1,q]+Log[1-q])/(Log[q]))+Sum[(((-1))^r)((3r-1)(q^(r(3r-1)/2))+3r (q^(r(3r+1)/2))),{r,150}],{q,0,20}]

CoefficientList[%,q]

(∑_n^∞)(((-1))^n) ((q^(n (3 n-1)/2)) (x^(3 n-1))+(q^(n (3 n+1)/2)) (x^(3 n)))+1==(∑_(n=0)^∞)(x^n) (QPochhammer[x, q, n+1])

f[x_,q_,inf:100]:=1+Sum[(((-1))^n)((x^(3n-1))(q^(n(3n-1)/2))+(x^(3n))(q^(n(3n+1)/2))),{n,inf}]

Series[f[x,q],{q,0,20}]

Expand/@Series[Sum[QPochhammer[{x,q},n+1](x^n),{n,0,inf}]/.inf->100,{q,0,20}]/.(x^(n_/;n>100)):>0

Series[1-Sum[QPochhammer[{x q,q},n-1](x^(n+1))(q^n),{n,inf}]/.inf->100,{q,0,20}]

Sum[(((-1))^n)((x^(3n-1))(q^(n(3n-1)/2))+(x^(3n))(q^(n(3n+1)/2))),{n,∞}]

ser1=1+Series[Sum[(((-1))^n)((x^(3n-1))(q^(n(3n-1)/2))+(x^(3n))(q^(n(3n+1)/2))),{n,100}],{x,0,10},{q,0,20}]

Sum[QPochhammer[x,q,n+1](x^n),{n,0,∞}]

ser2=Series[Sum[QPochhammer[x,q,n+1](x^n),{n,0,100}],{x,0,10},{q,0,20}]

ser1-ser2

Plot

With[{inf=100},Plot[Sum[QSeries[{q},inf]-QSeries[{q},n],{n,inf}],{q,0,1},PlotStyle->Red]
]//Timing

Block[{inf=100},ContourPlot[If[x^2+y^2>1,0,Re[Sum[QSeries[x+I y,inf]-QSeries[x+I y,n],{n,inf}]]],{x,-1,1},{y,-1,1},PlotPoints->200,ContourLines->False,ColorFunction->Hue]
]//Timing