Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots

实有理函数与Hermite-Biehler多项式的极大单叶圆盘

Maximal univalent disks of real rational functions and Hermite-Biehler polynomials

本文基于Hermite-Biehler定理，证明了以多项式p的相邻实根为直径的开圆盘是实有理函数R=q/p的极大单叶圆盘，从而解决了Hermite-Biehler问题的一个特例。

An estimate of the root mean square error incurred when approximating an $f \in L^2({\mathbb{R}})$ by a partial sum of its Hermite series

Let $f$ be a band-limited function in $L^2({\mathbb{R}})$. Fix $T >0$ and suppose $f^{\prime}$ exists and is integrable on $[-T, T]$. This paper gives a concrete estimate of the error incurred when approximating $f$ in the root mean square by a partial sum of its Hermite series. Specifically, we show, for $K=2n, \quad n \in Z_+,$ $$ \left[\frac{1}{2T}\int_{-T}^T[f(t)-(S_Kf)(t)]^2dt\right]^{1/2}\leq \left(1+\frac 1K\right)\left(\left[ \frac{1}{2T}\int_{|t|> T}f(t)^2dt\right]^{1/2} +\left[\frac{1}{2T} \int_{|ω|>N}|\hat f(ω)|^2dω\right]^{1/2} \right) +\frac{1}{K}\left[\frac{1}{2T}\int_{|t|\leq T}…

关于多元对称 Hermite 插值的一个注记

A note on the multivariate symmetric Hermite Interpolant

本文推广了多元对称多项式的 Hermite 插值，处理根重合情形，扩展了 Roy 和 Szpirglas 的对称 Hermite 插值基结果。

Supersymmetric Quantum mechanics on the radial lines

We investigate a type of Hermite orthogonal polynomials on $r$ lines in the plane which have a common point at the origin and endpoints at the $r$ roots of unity and we show that their related Hermite functions are eigenfunctions of a differential-difference operator. A supersymmetric harmonic oscillator on $r$ radial lines is presented and analyzed. Its eigenfunctions are given in terms of these polynomials.

A deformation of Hermite polynomials

We propose and study the properties of a set of polynomials $M_{nα, H\ }^{s}(z)$, $C_{nα, H}^{s}(z)$ $W_{nα, H}^{s}(z)$ with $n,s\in N$ $;α=\pm 1;$and where $H$ stands for Hermite ; the ''root '' polynomial >.These polynomials are obtained from a deformation of Hermite polynomials $H_{n}(z).$The structure underlying the deformation seems quite general and not only restricted to Hermite polynomials.

On a q-extension of Mehta's eigenvectors of the finite Fourier transform for q a root of unity

It is shown that the continuous q-Hermite polynomials for q a root of unity have simple transformation properties with respect to the classical Fourier transform. This result is then used to construct q-extended eigenvectors of the finite Fourier transform in terms of these polynomials.

Poles of Painlevé IV Rationals and their Distribution

We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite $H_{m,n}$ and generalised Okamoto $Q_{m,n}$ polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities…

Curie–Weiss铁磁体的李-杨零点、酉Hermite多项式与逆向热流

Lee--Yang zeroes of the Curie--Weiss ferromagnet, unitary Hermite polynomials, and the backward heat flow

本文研究逆向热流导出的酉Hermite多项式，将其零点分布识别为自由酉正态分布，并据此确定Curie–Weiss模型在复外场下的李-杨零点全局分布，同时揭示逆向热流在根分布层面诱导自由Brownian运动。

Distances of roots of classical orthogonal polynomials

Let $(P_N)_{N\ge0}$ one of the classical sequences of orthogonal polynomials, i.e., Hermite, Laguerre or Jacobi polynomials. For the roots $z_{1,N},\ldots, z_{N,N}$ of $P_N$ we derive lower estimates for $\min_{i\ne j}|z_{i,N}-z_{j,N}|$ and the distances from the boundary of the orthogonality intervals. The proofs are based on recent results on the eigenvalues of the covariance matrices in central limit theorems for associated $β$-random matrix ensembles where these entities appear as entries, and where the eigenvalues of these matrices are known.

A Semicircle Law for Derivatives of Random Polynomials

Let $x_1, \dots, x_n$ be $n$ independent and identically distributed random variables with mean zero, unit variance, and finite moments of all remaining orders. We study the random polynomial $p_n$ having roots at $x_1, \dots, x_n$. We prove that for $\ell \in \mathbb{N}$ fixed as $n \rightarrow \infty$, the $(n-\ell)-$th derivative of $p_n^{}$ behaves like a Hermite polynomial: for $x$ in a compact interval,$${n^{\ell/2}} \frac{\ell!}{n!} \cdot p_n^{(n-\ell)}\left( \frac{x}{\sqrt{n}}\right) \rightarrow He_{\ell}(x + γ_n),$$ where $He_{\ell}$ is the $\ell-$th probabilists' Hermite polynomial a…

实根保持线性变换的微分算子表示中系数多项式的交错性质

Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations

本文研究由实正交多项式系定义的线性变换，通过其微分算子表示中的系数多项式根的交错性质，刻画了保持实根性的条件，并证明相关猜想。

The quadratic formula made hard: A less radical approach to solving equations

It appears that, along with many of my friends and colleagues, I had been brainwashed by the great and tragic lives of Abel and Galois to believe that no general formulas are possible for roots of equations higher than quartic. This seemed to be confirmed by the brilliant and arduous solution of the general quintic by Hermite. Yet, below we find a formula giving a root to any algebraic equation of degree 2-5 and any reduced equation (see below) of higher degree. This algorithm, which must have been familiar to Lagrange, resulted when I was working on a paper on the asymptotics of hypergeometri…

Spherical transform and Jacobi polynomials on root systems of type BC

Let $R$ be a root system of type BC in $\mathfrak a=\mathbb R^r$ of general positive multiplicity. We introduce certain canonical weight function on $\mathbb R^r$ which in the case of symmetric domains corresponds to the integral kernel of the Berezin transform. We compute its spherical transform and prove certain Bernstein-Sato type formula. This generalizes earlier work of Unterberger-Upmeier, van Dijk-Pevsner, Neretin and the author. Associated to the weight functions there are Heckman-Opdam orthogonal polynomials of Jacobi type on the compact torus, after a change of variables they form an…

Counting monster potentials

We study the large momentum limit of the monster potentials of Bazhanov-Lukyanov-Zamolodchikov, which -- according to the ODE/IM correspondence -- should correspond to excited states of the Quantum KdV model. We prove that the poles of these potentials asymptotically condensate about the complex equilibria of the ground state potential, and we express the leading correction to such asymptotics in terms of the roots of Wronskians of Hermite polynomials. This allows us to associate to each partition of $N$ a unique monster potential with $N$ roots, of which we compute the spectrum. As a conseque…