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

Abstract

Let 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 p(n − 1)p(n + 1)/p(n) 2 upto a certain order so that one can finally ends up with the phenomena of 2-log-concavity and higher order Turán property of p(n) by following a sort of unified approach.