# Asymptotics of Certain q-Series

## TL;DR

This paper derives complete asymptotic expansions for specific q-series as q approaches 1 from below, revealing detailed behavior of these series in the scale of powers of log q.

## Contribution

It provides new asymptotic formulas for two classes of q-series involving divisor functions and power sums, extending understanding of their behavior near q=1.

## Key findings

- Asymptotic expansions in powers of log q as q approaches 1
- Explicit formulas for series involving divisor functions
- Enhanced understanding of q-series behavior near the singularity

## 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 $\sigma_{\alpha}(n)=\sum_{d\vert n}d^{\alpha}$.