A note on the higher order Turán inequalities for k-regular partitions

Abstract

Nicolas [8] and DeSalvo and Pak [3] proved that the partition function p(n) is log concave for $n \ge 25$. Chen et al. [2] proved that p(n) satisfies the third order Turán inequality, and that the associated degree 3 Jensen polynomials are hyperbolic for $n \ge 95$. Recently, Griffin et al. [5] proved more generally that for all d, the degree d Jensen polynomials associated to p(n) are hyperbolic for sufficiently large n. In this paper, we prove that the same result holds for the k-regular partition function $p_k(n)$ for $k \ge 2$. In particular, for any positive integers d and k, the order d Turán inequalities hold for $p_k(n)$ for sufficiently large n. The case when $d = k = 2$ proves a conjecture by Neil Sloane that $p_2(n)$ is log concave.