Lambert series and q-functions near q=1

Abstract

We study the Lambert series $\mathscr{L}_q(s,x) = \sum_{k=1}^\infty k^s q^{k x}/(1-q^k)$, for all $s \in \mathbb{C}$. We obtain the complete asymptotic expansion of $\mathscr{L}_q(s,x)$ 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.