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

Abstract

AbstractLet $${\overline{p}}(n)$$ 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., $$(-1)^{r-1}\Delta ^r \log {\overline{p}}(n)$$ ( - 1 ) r - 1 Δ r log p ¯ ( n ) , by studying the inequality of the following form $$\begin{aligned} \log \Bigl (1+\dfrac{C(r)}{n^{r-1/2}}-\dfrac{1+C_1(r)}{n^{r}}\Bigr ){} & {} <(-1)^{r-1}\Delta ^r \log {\overline{p}}(n) \\ {}{} & {} <\log \Bigl (1+\dfrac{C(r)}{n^{r-1/2}}\Bigr )\ \text {for}\ n \ge N(r), \end{aligned}$$ log ( 1 + C ( r ) n r - 1 / 2 - 1 + C 1 ( r ) n r ) < ( - 1 ) r - 1 Δ r log p ¯ ( n ) < log ( 1 + C ( r ) n r - 1 / 2 ) for n ≥ N ( r ) , where $$C(r), C_1(r), \text {and}\ N(r)$$ C ( r ) , C 1 ( r ) , 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 $$(-1)^{r-1}\Delta ^r \log {\overline{p}}(n)$$ ( - 1 ) r - 1 Δ r log p ¯ ( n ) than 0. By settling the problem, we are able to show that $$\begin{aligned} \lim _{n\rightarrow \infty }(-1)^{r-1}\Delta ^r \log {\overline{p}}(n) =\dfrac{\pi }{2}\Bigl (\dfrac{1}{2}\Bigr )_{r-1}n^{\frac{1}{2}-r}. \end{aligned}$$ lim n → ∞ ( - 1 ) r - 1 Δ r log p ¯ ( n ) = π 2 ( 1 2 ) r - 1 n 1 2 - r .