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

Ken Ono

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 Turán 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 $|m-μ|< Cσ,$ where $μ$ and $σ$ 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.

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

Abstract

In 1878, Sylvester proved Cayley's Conjecture that the coefficients of the Gaussian -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 log-concavity, known as Turán inequalities. Although -binomial coefficients are not always log- or degree 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 on the central window where and are the mean and standard deviation of the normalized distribution. More generally, we obtain the same conclusions for -multinomial coefficients. These results stem from the asymptotic behavior of normalized Jensen polynomials, which are approximated by Hermite polynomials.

Paper Structure

This paper contains 8 sections, 9 theorems, 81 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Nuts and bolts and the proof of Theorem \ref{['thm:HermiteLimit']} and Corollary \ref{['cor:hyperTuran']}
  3. Nuts and Bolts
  4. Proof of Theorem \ref{['thm:HermiteLimit']}
  5. Proof of Corollary \ref{['cor:hyperTuran']}
  6. Proof of Theorem \ref{['thm:multinom-hermite']} and Corollary \ref{['cor:multinom-turan']}
  7. Sketch of the proof of Theorem \ref{['thm:multinom-hermite']}
  8. Sketch of the proof of Corollary \ref{['cor:multinom-turan']}

Key Result

Theorem 1

Fix $d\ge1$ and $\lambda\in(0,1)$, and suppose $a,b\to +\infty$ with $a/(a+b)\to\lambda$. Then for every $C>0$, uniformly for integers $m$ with $|m-\mu_{a,b}|\le C\,\sigma_{a,b}$, coefficientwise we have

Theorems & Definitions (19)

  • Theorem 1
  • Example
  • Corollary 2
  • Theorem 3
  • Remark
  • Example
  • Corollary 4
  • Lemma 5: Mean and variance
  • proof
  • Lemma 6: log-ratio in the central window
  • ...and 9 more