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

TL;DR

This paper investigates the higher order log-concavity properties of the overpartition function, extending previous work on partition functions, and establishes new results on its asymptotic behavior and Turán-type inequalities.

Contribution

It introduces the study of higher order log-concavity for overpartition functions and proves their asymptotic properties and Turán inequalities using a unified approach.

Findings

Proves higher order log-concavity of overpartition function asymptotically.

Establishes 2-log-concavity and Turán inequalities for overpartitions.

Extends methods from partition functions to overpartitions.

Abstract

Let $\bar{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher order $\log$-concavity property of the overpatition 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 $\bar{p}(n-1)\bar{p}(n+1)/\bar{p}(n)^2$ upto a certain order so that one can finally ends up with the phenomena of $2$-$\log$-concavity and higher order Tur\'{a}n property of $\bar{p}(n)$ by following a sort of unified approach.