Higher order Turán inequalities for the partition function

By William Y. C. Chen, Dennis X. Q. Jia, Larry X. W. Wang. Transactions of the American Mathematical Society, 2017

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The Turan inequalities and the higher order Turan inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Polya class. A real sequence $\{a_{n}\}$ is said to satisfy the Turan inequalities if for $n\geq 1$, $a_n^2-a_{n-1}a_{n+1}\geq 0$. It is said to satisfy the higher order Turan inequalities if for $n\geq 1$, $4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq 0$. A sequence satisfying the Turan inequalities is also called log-concave. For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, the sequence $\{ p(n)\}_{n> 25}$ is log-concave, that is, $p(n)^2-p(n-1)p(n+1)>0$ for $n> 25$. It was conjectured by Chen that $p(n)$ satisfies the higher order Turan inequalities for $n\geq 95$. In this paper, we prove this conjecture by using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and a lower bound for $p(n+1)p(n-1)/p(n)^2$. Consequently, for $n\geq 95$, the Jensen polynomials $g_{3,n-1}(x)=p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3$ have only real zeros. We conjecture that for any positive integer $m\geq 4$ there exists an integer $N(m)$ such that for $n\geq N(m) $, the polynomials $\sum_{k=0}^m {m\choose k}p(n+k)x^k$ have only real zeros. This conjecture was independently posed by Ono.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/c359f618669d21d8f4cb76642e2a7fa9c66ee3b4