r-log-concavity of partition functions

Abstract

Let $\hat{\mathscr {L}}$ be the operator given by $\hat{\mathscr {L}} \{a_n\}_{n \ge 0} = \{a_{n+1}^2 - a_{n} a_{n+2} \}_{n \ge 0}$. A sequence $\{ a_n \}_{n \ge 0}$ is called asymptotically r-log-concave if $\hat{\mathscr {L}}^k \{a_n\}_{n \ge N}$ are non-negative sequences for $1 \le k \le r$ and some integer N. Let p(n) be the number of integer partitions of n. We prove that the sequence $\{p(n)\}_{n \ge 1}$ is asymptotically r-log-concave for any positive integer r. Moreover, we give a method to compute the explicit N such that $\{p(n)\}_{n \ge N}$ is r-log-concave.