q-Harmonic Series

Author

Eric W. Weisstein
December 7, 2008

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

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

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

Code

HarmonicH[r_,q_]:=Sum[1/(q^k-r),{k,Infinity}]
HarmonicLn[r_,q_]:=Sum[(-1)^n/(q^n+(1-r)),{n,Infinity}]

Numeric Values

Table[N[h[i,1],10],{i,2,5}]

Table[N[Ln[i,2],10],{i,2,5}]

Sums

HarmonicH[-r,q]

HarmonicLn[1-r,q]

Special Cases

HarmonicH[1,q]

HarmonicH[1,q]

Sum[((q^n)/((1-(q^n))QPochhammer[q,q,n])),{n,∞}]//Timing

TimeConstrained[Sum[(1-(q^n)-(q^(2n)))/(((q^n)-1)QBinomial[2n,n,q]QPochhammer[q,q,n]),{n,∞}],120]//Timing

HarmonicLn[2,q]

HarmonicLn[2,q]

TimeConstrained[Sum[(((q^n)QPochhammer[q,q,n])/((1-(q^n))QPochhammer[(q^2),(q^2),n])),{n,∞}],120]//Timing

TimeConstrained[Sum[(((((-1))^(n-1))QPochhammer[q,q,n](1-(q^(3n))))/((((1-(q^n)))^2)Binomial[2n,n]QPochhammer[(q^2),(q^2),n])),{n,∞}],120]//Timing