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

Abstract

Let $${\overline{p}}(n)$$ 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 .

Restricted partition functions and the r-log-concavity of quasi-polynomial-like functions

“…It is worth noting that despite the fact that the former proof requires more sophisticated preparation than the latter one, the method which we use there might be also effectively applied for more complicated functions like the partition function [22] or the overpartition function [28].…”

“…Moreover, these authors used that result and proved that the partition function p(n) is (asymptotically) r-logconcave for each r ∈ N + , for more details see [22]. Afterwards, Mukherjee, Zhang and Zhong [28] applied the aforementioned criterion and showed that the overpartition function is (asymptotically) r-log-concave for any r ∈ N + . Recall that an overpartition [9] of an integer n is a partition of n where the őrst occurrence of every distinct part may be overlined.…”

Higher order log-concavity of the overpartition function and its consequences

“…They also proved the higher-order Turán property of for (see [16, Theorem 1.2]). Following the treatment in [12], the first author [17, Theorem 1.7] laid out a proof that is 2-log-concave.…”