Asymptotics of Certain q-Series

Abstract

In this work we study complete asymptotic expansions for the q-series $\sum_{n=1}^{\infty}\frac{1}{n^{b}}q^{n^{a}}$ and $\sum_{n=1}^{\infty}\frac{\sigma_{\alpha}(n)}{n^{b}}q^{n^{a}}$ in the scale function $(\log q)^{n}$ as $q\to1^{-}$, where $a>0,\ q\in(0,1),\,b,\alpha\in\mathbb{C}$ and $\sigma_{\alpha}(n)$ is the divisor function $\sigma_{\alpha}(n)=\sum_{d\vert n}d^{\alpha}$.