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

1. Problem Statement and Context

The paper addresses the structural properties of the coefficients of Gaussian (q-binomial) coefficients and their generalization, q-multinomial coefficients.

• Historical Background: In 1856, Cayley conjectured that the coefficients of the q-binomial coefficient $\binom{a+b}{a}_q$ are unimodal. This was proven by Sylvester (1878) using invariant theory and later by O'Hara (1990) via a constructive combinatorial bijection. Pak and Panova (2013) further established strict unimodality for sufficiently large parameters.
• The Core Question: While unimodality is well-understood, the paper investigates stronger properties: log-concavity and higher-degree log-concavity (known as Turán inequalities).
  • A sequence $(a_k)$ is log-concave if $a_k^2 \geq a_{k-1}a_{k+1}$.
  • It is $d$-log-concave if the iterated operator $L^r a$ is non-negative for all $1 \leq r \leq d$, where $(La)_k = a_k^2 - a_{k-1}a_{k+1}$.
• The Challenge: q-binomial coefficients are not globally log-concave (or $d$-log-concave) for all indices. For example, the coefficients of $\binom{4}{2}_q = 1 + q + 2q^2 + q^3 + q^4$ violate log-concavity at $k=1$ ($1^2 < 1 \cdot 2$).
• Objective: The paper seeks to determine the extent to which these stronger inequalities hold. Specifically, it asks if log-concavity holds within a "central window" of the coefficients when the parameters $a$ and $b$ are large and balanced.

2. Methodology

The author employs a sophisticated combination of probabilistic limit theorems, asymptotic analysis, and the theory of Jensen polynomials.

A. Probabilistic Framework

The coefficients $c_{a,b}(k)$ of the q-binomial coefficient are normalized to form a probability distribution $p_{a,b}(k)$.

• Mean ($\mu$) and Variance ($\sigma^2$): The paper establishes that for the random variable $K$ representing the index $k$, the mean is $\mu_{a,b} = ab/2$ and the variance is $\sigma^2_{a,b} = ab(a+b+1)/12$.
• Central Window: The analysis focuses on indices $m$ within a window of size $C\sigma$ around the mean: $|m - \mu| \leq C\sigma$.

B. Jensen Polynomials and Hermite Limits

The core technical tool is the Jensen polynomial of degree $d$ at index $m$:
$$J_{d,m}(X; u) = \sum_{j=0}^d \binom{d}{j} u_{m+j} X^j$$
The paper defines normalized Jensen polynomials $J^{d,m}_{a,b}(X)$ by scaling the variable and coefficients using the standard deviation $\sigma$ and a scaling factor $\delta = 1/\sqrt{2}\sigma$.

Key Strategy:

1. Quadratic Log-Ratio Model: Using Edgeworth expansions and cumulant analysis, the author proves that within the central window, the log-ratio of consecutive coefficients behaves quadratically:
   $$\log \frac{p_{a,b}(m+j)}{p_{a,b}(m)} = A j - \delta^2 j^2 + O((a+b)^{-1/2})$$
   Crucially, the third cumulant $\kappa_3$ vanishes due to the symmetry (palindromicity) of the distribution.
2. Convergence to Hermite Polynomials: By substituting this quadratic model into the definition of the normalized Jensen polynomials, the author demonstrates that as $a, b \to \infty$ (with a fixed aspect ratio), these polynomials converge coefficientwise to the physicists' Hermite polynomials $H_d(X)$:
   $$J^{d,m}_{a,b}(X) = H_d(X) + O((a+b)^{-1/2})$$
3. Hurwitz Continuity: Since Hermite polynomials $H_d(X)$ are known to be hyperbolic (all roots are real and simple), the convergence implies that for sufficiently large parameters, the Jensen polynomials of the q-binomial coefficients are also real-rooted.

C. Connection to Turán Inequalities

The paper utilizes a theorem by Craven and Csordas, which states that if the Jensen polynomials of a sequence are real-rooted, then the sequence satisfies the corresponding Turán inequalities (degree $d$ log-concavity).

3. Key Contributions and Results

Main Theorems

• Theorem 1 (q-Binomials): For fixed degree $d \geq 1$ and aspect ratio $\lambda \in (0,1)$, as $a, b \to \infty$ with $a/(a+b) \to \lambda$, the normalized Jensen polynomials for indices in the central window $|m - \mu| \leq C\sigma$ converge uniformly to the Hermite polynomial $H_d(X)$.
• Corollary 2 (Consequences for q-Binomials):
  1. Hyperbolicity: For sufficiently large balanced $a, b$, the normalized Jensen polynomials are real-rooted.
  2. d-Log-Concavity: The coefficients $c_{a,b}(k)$ are $d$-log-concave for all indices $k$ in the central window $W_{a,b}$.
• Theorem 3 & Corollary 4 (q-Multinomials): These results are generalized to q-multinomial coefficients $\binom{n}{n_1, \dots, n_r}_q$. Under similar conditions (fixed number of parts $r$, proportions bounded away from 0 and 1), the coefficients are $d$-log-concave in the central window defined by the multinomial mean and variance.

Technical Innovations

• Uniformity: The results hold uniformly for any fixed degree $d$ and any window size $C$, provided the parameters are large enough.
• Explicit Error Bounds: The convergence rate is explicitly quantified as $O((a+b)^{-1/2})$ for binomials and $O(n^{-1/2})$ for multinomials.
• Cumulant Analysis: The proof rigorously handles the vanishing of odd cumulants (due to symmetry) and bounds the fourth cumulant to ensure the quadratic approximation holds.

4. Significance

• Resolution of a Local Property: While global log-concavity fails for q-binomials, this paper proves that the "strongest" form of unimodality (infinite log-concavity) holds locally in the central region of the distribution.
• Bridge Between Combinatorics and Analysis: The work connects combinatorial partition theory (q-analogs) with the analytic theory of polynomials (Jensen and Hermite limits), a strategy previously applied to the Riemann Hypothesis (Griffin, Ono, Rolen, Zagier).
• Generalization: By extending the results from q-binomials to q-multinomials, the paper provides a unified framework for understanding the asymptotic shape of coefficients in a broad class of q-series.
• Methodological Impact: The use of Edgeworth expansions to derive quadratic log-ratio models for discrete distributions offers a powerful template for analyzing the local limit behavior of other combinatorial sequences.

In summary, Ken Ono demonstrates that while q-multinomial coefficients are not globally log-concave, they exhibit asymptotic $d$-log-concavity in their central regions, driven by their convergence to Hermite polynomials under appropriate normalization.