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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 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.

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Authors' contributions

We would like to thank Ken Ono for suggesting this problem and his guidance.

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Correspondence to William Craig.

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We acknowledge the support of NSF Grant # DMS-1601306.

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Craig, W., Pun, A. A note on the higher order Turán inequalities for k-regular partitions. Res. number theory 7, 5 (2021).

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