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

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 Turan 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 Turan 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.