Hermite-Jensen limits and $d$ log-concavity of $q$-multinomials

TL;DR

This paper investigates the extent of log-concavity and higher degree Turán inequalities for $q$-multinomials, demonstrating uniform validity in certain regimes via asymptotic analysis of Jensen polynomials approximated by Hermite polynomials.

Contribution

It extends the understanding of log-concavity properties from $q$-binomials to $q$-multinomials, establishing uniform inequalities in specific asymptotic regimes.

Findings

Stronger inequalities hold uniformly in the central window for $q$-multinomials.

Asymptotic behavior of Jensen polynomials approximates Hermite polynomials.

Results apply to families with limiting aspect ratio bounded away from zero and one.

Abstract

In 1878, Sylvester proved Cayley's Conjecture that the coefficients of the Gaussian $q$-binomial coefficients are unimodal. In 1990, O'Hara famously discovered a constructive combinatorial proof, and in 2013, Pak and Panova proved the stronger property of strict unimodality for sufficiently large parameters. We move from unimodality to log-concavity and higher degree $ d$ log-concavity, known as Tur\'an inequalities. Although $q$-binomial coefficients are not always log- or degree $d$ log-concave, it's natural to ask to what extent these inequalities hold. In infinite families with limiting aspect ratio bounded away from zero and one, we prove that these stronger inequalities hold uniformly, for each $C>0,$ on the central window $|m-\mu|< C\sigma,$ where $\mu$ and $\sigma$ are the mean and standard deviation of the normalized distribution. More generally, we obtain the same conclusions for $q$-multinomial coefficients. These results stem from the asymptotic behavior of normalized Jensen polynomials, which are approximated by Hermite polynomials.