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

Abstract

Let ${\overline{p}}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher-order log-concavity property of the overpartition function in a similar framework done by Hou and Zhang for the partition function. This will enable us to move on further in order to prove log-concavity of overpartitions, explicitly by studying the asymptotic expansion of the quotient ${\overline{p}}(n-1){\overline{p}}(n+1)/{\overline{p}}(n)^2$ up to a certain order. This enables us to additionally prove 2-log-concavity and higher Turán inequalities with a unified approach.

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.…”