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 $q\to 1^{-}$, where $a>0,q\in (0,1),b,\alpha \in \mathbb{C}$ and ${\sigma }_{\alpha }(n)$ is the divisor function ${\sigma }_{\alpha }(n)={\sum }_{d|n}{d}^{\alpha }$.