Interlacing Log-concavity of the Boros-Moll Polynomials

TL;DR

This paper introduces the concept of interlacing log-concavity for polynomial sequences, proves that Boros-Moll polynomials possess this property, and provides conditions under which classical combinatorial polynomials are also interlacing log-concave.

Contribution

The paper defines interlacing log-concavity, proves Boros-Moll polynomials have this property, and offers sufficient conditions for other polynomials to be interlacing log-concave.

Findings

Boros-Moll polynomials are interlacing log-concave.

Interlacing log-concavity is a stronger property than log-concavity.

Sufficient conditions are provided for classical combinatorial polynomials to be interlacing log-concave.

Abstract

We introduce the notion of interlacing log-concavity of a polynomial sequence $\{P_m(x)\}_{m\geq 0}$, where $P_m(x)$ is a polynomial of degree m with positive coefficients $a_{i}(m)$. This sequence of polynomials is said to be interlacing log-concave if the ratios of consecutive coefficients of $P_m(x)$ interlace the ratios of consecutive coefficients of $P_{m+1}(x)$ for any $m\geq 0$. Interlacing log-concavity is stronger than the log-concavity. We show that the Boros-Moll polynomials are interlacing log-concave. Furthermore we give a sufficient condition for interlacing log-concavity which implies that some classical combinatorial polynomials are interlacing log-concave.