On asymptotic formulas for certain $q$-series involving partial theta functions

Abstract:

In this article, we investigate the integer sequence $a_d (n)$, where for a positive integer $d$, $a_d (n)$ is defined by \[ \prod _{n=1}^{\infty } \frac {1}{1-q^n} \sum _{n=0}^{\infty } (-1)^n q^{ ( d n^2 - ( d - 2) n ) / 2 } = \sum _{n=0}^{\infty } a_d (n) q^n . \] In particular, we will present a combinatorial model for $a_d (n)$ via integer partitions and this will play a crucial role in obtaining an asymptotic formula for $a_d (n)$. Moreover, we will show that $p(n) - 2 a_d (n)$ has an unexpected sign pattern via combinatorial arguments and asymptotic formulas.