Inequalities for higher order differences of the logarithm of the overpartition function and a problem of Wang-Xie-Zhang.

Abstract

Let $\overline{p}\left(n\right)$ denote the overpartition function. In this paper, our primary goal is to study the asymptotic behavior of the finite differences of the logarithm of the overpartition function, i.e., ${(-1)}^{r-1}{\Delta }^{r}log\overline{p}\left(n\right)$ , by studying the inequality of the following form $\begin{array}{cc}log(1+\frac{C\left(r\right)}{n^{r-1/2}}-\frac{1+C_1\left(r\right)}{n^r})& <{(-1)}^{r-1}{\Delta }^{r}log\overline{p}\left(n\right)\\ & <log(1+\frac{C\left(r\right)}{n^{r-1/2}})\text{for}n\ge N\left(r\right),\end{array}$ where $C\left(r\right),C_1\left(r\right),\text{and}N\left(r\right)$ are computable constants depending on the positive integer r, determined explicitly. This solves a problem posed by Wang, Xie and Zhang in the context of searching for a better lower bound of ${(-1)}^{r-1}{\Delta }^{r}log\overline{p}\left(n\right)$ than 0. By settling the problem, we are able to show that $\begin{array}{c}\underset{n\to \infty }{lim}{(-1)}^{r-1}{\Delta }^{r}log\overline{p}\left(n\right)=\frac{\pi }{2}(\frac{1}{2}{)}_{r-1}{n}^{\frac{1}{2}-r}.\end{array}$.