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

Abstract

Let $\overline{p}(n)$ 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., $(-1)^{r-1}\Delta^r \log \p(n)$, by studying the inequality of the following form $$\log \Bigl(1+\dfrac{C(r)}{n^{r-1/2}}-\dfrac{C_1(r)}{n^{r}}\Bigr)<(-1)^{r-1}\Delta^r \log \p(n) <\log \Bigl(1+\dfrac{C(r)}{n^{r-1/2}}\Bigr)\ \text{for}\ n \geq N(r),$$ where $C(r), C_1(r), \text{and}\ N(r)$ 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 $(-1)^{r-1}\Delta^r \log \p(n)$ than $0$. By settling the problem, we are able to show that \begin{equation*} \lim_{n\rightarrow \infty}(-1)^{r-1}\Delta^r \log \p(n) =\dfrac{\pi}{2}\Bigl(\dfrac{1}{2}\Bigr)_{r-1}n^{\frac{1}{2}-r}. \end{equation*}