Lambert series and q-functions near q=1

Abstract

We study the Lambert series ${\mathcal{L}}_q(s,x)={\sum }_{k=1}^{\infty }{k}^s{q}^{kx}/(1-q^k)$, for all $s\in \mathbb{C}$. We obtain the complete asymptotic expansion of ${\mathcal{L}}_q(s,x)$ near $q=1$. Our analysis of the Lambert series yields the asymptotic forms for several related q-functions: the q-gamma and q-polygamma functions, the q-Pochhammer symbol, and, in closed form, the Jacobi theta functions. Some typical results include ${\Gamma }_2(\frac{1}{4}){\Gamma }_2(\frac{3}{4})\simeq \frac{2^{13/32}\pi }{\log 2}$ and ${\vartheta }_4(0,e^{-1/\pi })\simeq 2\pi e^{-{\pi }^3/4}$, with relative errors of order $10^{-25}$ and $10^{-27}$ respectively.

Submitted for publication

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