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 $\sum_{n=1}^{\infty}\frac{σ_α(n)}{n^{b}}q^{n^{a}}$ in the scale function $(\log q)^{n}$ as $q\to1^{-}$, where $a>0,\ q\in(0,1),\,b,α\in\mathbb{C}$ and $σ_α(n)$ is the divisor function $σ_α(n)=\sum_{d\vert n}d^α$.