Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.01456 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908752156819456 |
|---|---|
| author | Kabluchko, Zakhar |
| author_facet | Kabluchko, Zakhar |
| contents | 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 $L_n^*(x; nβ, \lfloor n γ\rfloor)$ converges to the free multiplicative Poisson distribution on the positive half-line (when $γ>0$ and $β\in \mathbb R\backslash[0,1]$) or on the unit circle (when $γ>0$ and $β\in -\frac 12 + \sqrt{-1} \, \mathbb R$). 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zero distribution of multiplicative Hermite and Laguerre polynomials Kabluchko, Zakhar Probability Classical Analysis and ODEs Primary: 46L54, 26C10. Secondary: 30C15, 33C45, 30C10, 60B10, 34A99, 35F99 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 $L_n^*(x; nβ, \lfloor n γ\rfloor)$ converges to the free multiplicative Poisson distribution on the positive half-line (when $γ>0$ and $β\in \mathbb R\backslash[0,1]$) or on the unit circle (when $γ>0$ and $β\in -\frac 12 + \sqrt{-1} \, \mathbb R$). 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. |
| title | Zero distribution of multiplicative Hermite and Laguerre polynomials |
| topic | Probability Classical Analysis and ODEs Primary: 46L54, 26C10. Secondary: 30C15, 33C45, 30C10, 60B10, 34A99, 35F99 |
| url | https://arxiv.org/abs/2511.01456 |