Asymptotics of the shifted finite differences of the overpatition function and a problem of Wang--Xie--Zhang

Abstract

Let $\overline{p}(n)$ denote the overpartition function, and for $j\in \mathbb{N}$, ${\Delta }_j^r$ denote the $r$-fold applications of the shifted difference operator ${\Delta }_j$ defined by ${\Delta }_j(a)(n):=a(n)-a(n-j)$. The main goal of this paper is to derive an asymptotic expansion of ${\Delta }_j^r(\overline{p})(n)$ with an effective error bound which subsequently gives an answer to a problem of Wang, Xie, and Zhang. In order to get the asymptotics of ${\Delta }_j^r(\overline{p})(n)$, we derive an asymptotic expansion of the shifted overpartition function $\overline{p}(n+k)$ for any integer $k\ne 0$.