q-Polygamma Function

Author

Eric W. Weisstein
December 29, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/q-PolygammaFunction.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Usage

?QGamma

Identities

(QPolyGamma3[0, z, q])==(QPolyGamma[z, q])

QPolyGamma[z,q]

Definition

((D[QGamma[z,q],z])/(QGamma[z,q]))

Integral

Integrate[QPolyGamma[z,q],z]

Integrate[QPolyGamma[1,z,q],z]

Integrate[D[QPolyGamma[z,q],z],z]

D[Integrate[QPolyGamma[z,q],z],z]

Derivative

Table[D[QPolyGamma[z,q],{z,n}],{n,0,3}]

Sums

L(q)==(∑_n^∞)((q^n)/(1-(q^n)))==(((QPolyGamma[1, q])+log(1-q))/(log(q)))

L[q_,z_]:=((QPolyGamma[z,q]+Log[1-q])/(Log[q]))

Sum[((b^n)/(1-(b^n))),{n,∞}]

Sum[((b^n)/(1-(b^n))),n]

Sum[((b^n)/(1-(b^n))),{n,k}]

Limit[%,k->0]

Limit[%,k->∞]

Series[((QPolyGamma[0,1+k,b])/(Log[b])),{k,∞,3}]

((QPolyGamma[0,1+k,b])/(Log[b]))/.b->1/3/.k->999999.

log(q) (∑_(n=0)^∞)((q^(n+z))/(1-(q^(n+z))))-log(1-q)==(QPolyGamma3[0, z, q])

V7.0.0

-Log[1-q]+Log[q]Sum[((q^(n+z))/(1-(q^(n+z)))),{n,0,∞}]

V7.0.1

-Log[1-q]+Log[q]Sum[((q^(n+z))/(1-(q^(n+z)))),{n,0,∞}]

(∑_(k=0)^∞)(1/(cosh(k)+1))==2-2 (QPolyGamma3[1, -i π, e])

Sum[(1/(Cosh[k]+1)),{k,0,Infinity}]

FullSimplify[%]

(∑_n^∞)(1/(1-(a^n)))==(((QPolyGamma3[0, 1, 1/a])+log(a-1)+log((1/a)))/(log(a)))

Sum[1/(1 - a^n), {n, Infinity}]

FullSimplify[%]

Identities

((EllipticTheta[2, 0, (ϕ^(-2))])^2)==-((π-i ((QPolyGamma3[0, (1/2)-((i π)/(4 log(ϕ))), (ϕ^2)])-(QPolyGamma3[0, (1/2)+((i π)/(4 log(ϕ))), (ϕ^2)])))/(log(ϕ)))

s1=-(1/(Log[GoldenRatio]))(π-I (QPolyGamma[0,(1/2)-((I π)/(4 Log[GoldenRatio])),(GoldenRatio^2)]-QPolyGamma[0,(1/2)+((I π)/(4 Log[GoldenRatio])),(GoldenRatio^2)]));

s2= ((EllipticTheta[2,0,(GoldenRatio^(-2))])^2);

N[s1-s2,1000]