A note on the Higher order Tur\'{a}n inequalities for $k$-regular partitions

TL;DR

This paper extends the understanding of Turán inequalities and hyperbolicity of Jensen polynomials from the partition function to the $k$-regular partition function, proving large $n$ behavior and confirming a conjecture for $k=2$.

Contribution

It proves that the degree $d$ Jensen polynomials associated with $p_k(n)$ are hyperbolic for large $n$, generalizing previous results to $k$-regular partitions.

Findings

Degree $d$ Jensen polynomials are hyperbolic for large $n$

Order $d$ Turán inequalities hold for $p_k(n)$ for large $n$

Confirms Sloane's conjecture for $p_2(n)$ being log concave

Abstract

Nicolas and DeSalvo and Pak proved that the partition function $p(n)$ is log concave for $n \geq 25$. Chen, Jia and Wang proved that $p(n)$ satisfies the third order Tur\'{a}n inequality, and that the associated degree 3 Jensen polynomials are hyperbolic for $n \geq 94$. 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 $p_k(n)$ for $k \geq 2$. In particular, for any positive integers $d$ and $k$, the order $d$ Tur\'{a}n 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.