Zero distribution of multiplicative Hermite and Laguerre polynomials

TL;DR

This paper investigates the asymptotic zero distributions of multiplicative Hermite and Laguerre polynomials, revealing their convergence to various free multiplicative distributions on the positive line or the unit circle.

Contribution

It introduces multiplicative analogues of Hermite and Laguerre polynomials and establishes their zero distribution convergence to free multiplicative laws, unifying different cases with a common method.

Findings

Zero distributions of multiplicative Hermite polynomials converge to free multiplicative normal laws.

Zero distributions of multiplicative Laguerre polynomials converge to free multiplicative Poisson laws.

Results unify Hermite and Laguerre cases in a common analytical framework.

Abstract

It is well-known that, as $n\to\infty$, the zero distribution of the $n$-th Hermite polynomial converges to the semicircular law (the free normal distribution), while the zero distribution of the associated Laguerre polynomials converges to the Marchenko--Pastur law (the free Poisson distribution). In this paper, we establish multiplicative analogues of these results. We define the multiplicative Hermite and Laguerre polynomials by \begin{align*} H_n^*(x;s) &:= e^{-\frac 12 s ((x\partial_x)^2 - n x \partial_x) } (x-1)^n = \sum_{j=0}^n (-1)^{n-j} \binom nj e^{-\frac 12 s (j^2 - nj)} x^j, \\ L_n^*(x; b,c) &:= (x\partial_x + b)^c (x-1)^n = \sum_{j=0}^n (-1)^{n-j} \binom nj (j+b)^c x^j, \end{align*} where $n\in \mathbb N_0$, $\partial_x$ denotes the differentiation operator w.r.t. $x$, and $s\in \mathbb R$, $b\in \mathbb C$, $c\in \mathbb N_0$ are parameters. In the Hermite case, we show that, as $n\to\infty$, the zero distribution of $H_n^*(x;s/n)$ converges weakly to the free multiplicative normal distribution on the positive half-line (when $s>0$) or to the free unitary normal distribution on the unit circle $\{|z| = 1\}$ (when $s<0$). In the Laguerre case, we show that the zero distribution of $L_n^*(x; n\beta, \lfloor n \gamma \rfloor)$ converges to the free multiplicative Poisson distribution on the positive half-line (when $\gamma >0$ and $\beta \in \mathbb R\backslash[0,1]$) or on the unit circle (when $\gamma>0$ and $\beta \in -\frac 12 + \sqrt{-1} \, \mathbb R$). All these results are obtained by essentially the same method, which treats the Hermite/Laguerre cases and the unitary/positive settings in a unified way.