Hahn polynomials and the Burnside process

多元Hahn多项式与差分方程

Multivariate Hahn polynomials and difference equations

本文显式构造多元Hahn多项式作为一族可交换二阶差分算子的公共特征向量，证明其关于超几何多项分布正交，并通过极限过程导出完备的多元Krawtchouk和Meixner多项式集。

Multivariate Jacobi and Laguerre polynomials, infinite-dimensional extensions, and their probabilistic connections with multivariate Hahn and Meixner polynomials

Multivariate versions of classical orthogonal polynomials such as Jacobi, Hahn, Laguerre and Meixner are reviewed and their connection explored by adopting a probabilistic approach. Hahn and Meixner polynomials are interpreted as posterior mixtures of Jacobi and Laguerre polynomials, respectively. By using known properties of gamma point processes and related transformations, a new infinite-dimensional version of Jacobi polynomials is constructed with respect to the size-biased version of the Poisson--Dirichlet weight measure and to the law of the gamma point process from which it is derived.

Non-intersecting paths and Hahn orthogonal polynomial ensemble

We compute the bulk limit of the correlation functions for the uniform measure on lozenge tilings of a hexagon. The limiting determinantal process is a translation invariant extension of the discrete sine process, which also describes the ergodic Gibbs measure of an appropriate slope.

Moments of Generalized Cauchy Random Matrices and continuous-Hahn Polynomials

In this paper we prove that, after an appropriate rescaling, the sum of moments $\mathbb{E}_{N}^{(s)} \left( Tr \left( |\mathbf{H}|^{2k+2}+|\mathbf{H}|^{2k}\right) \right)$ of an $N\times N$ Hermitian matrix $\mathbf{H}$ sampled according to the generalized Cauchy (also known as Hua-Pickrell) ensemble with parameter $s>0$ is a continuous-Hahn polynomial in the variable $k$. This completes the picture of the investigation that began by Cunden, Mezzadri, O'Connell and Simm who obtained analogous results for the other three classical ensembles of random matrices, the Gaussian, the Laguerre and Ja…

有限马尔可夫链与多重正交多项式

Finite Markov chains and multiple orthogonal polynomials

本文利用正交多项式与多重正交多项式的递推矩阵构建有限马尔可夫链，推导其谱理论与多项性质，揭示生灭过程因式分解特性，并结合超几何多项式实例验证，构建对应常返链与反转链。

对称近似有界不可区分性的上界

Symmetric Bounded Indistinguishability: Hypergeometric Smoothing and Hahn Polynomials

本文通过超几何平滑和汉恩多项式方法，获得更宽参数范围的上界，改进了多个领域内的已有结果，证明了在特定参数下，c2n阶边际统计距离必须指数级接近。