Séminaire Lotharingien de Combinatoire, 84B.21 (2020), 8 pp.
William Craig and Anna Ying Pun
Higher order Turán inequalities for k-regular partitions
Abstract.
Nicolas (in 1978)
and DeSalvo and Pak (in 2015)
proved that the partition function p(n) is log concave for
n >= 25. In 2019, Chen, Jia and Wang
proved that p(n) satisfies the third order Turán inequality, and that the associated degree 3 Jensen polynomials are hyperbolic for
n >= 94. More recently, Griffin, Ono, Rolen and Zagier
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 pk(n) for
k >= 2. In particular, for any positive integers d and k, the order d Turán inequalities hold for pk(n) for sufficiently large n. The case when d=k=2 proves a conjecture by Neil Sloane that p2(n) is log concave.
Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.
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