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

Laguerre and Meixner symmetric functions, and infinite-dimensional diffusion processes

The Laguerre symmetric functions introduced in the note are indexed by arbitrary partitions and depend on two continuous parameters. The top degree homogeneous component of every Laguerre symmetric function coincides with the Schur function with the same index. Thus, the Laguerre symmetric functions form a two-parameter family of inhomogeneous bases in the algebra of symmetric functions. These new symmetric functions are obtained from the N-variate symmetric polynomials of the same name by a procedure of analytic continuation. The Laguerre symmetric functions are eigenvectors of a second order…

Markov Chains Generated by Convolutions of Orthogonality Measures

About two dozens of exactly solvable Markov chains on one-dimensional finite and semi-infinite integer lattices are constructed in terms of convolutions of orthogonality measures of the Krawtchouk, Hahn, Meixner, Charlier, $q$-Hahn, $q$-Meixner and little $q$-Jacobi polynomials. By construction, the stationary probability distributions, the complete sets of eigenvalues and eigenvectors are provided by the polynomials and the orthogonality measures. An interesting property possessed by these stationary probability distributions, called `convolutional self-similarity,' is demonstrated.

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。

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…

Polynomials of Meixner's type in infinite dimensions-Jacobi fields and orthogonality measures

The classical polynomials of Meixner's type--Hermite, Charlier, Laguerre, Meixner, and Meixner--Pollaczek polynomials--are distinguished through a special form of their generating function, which involves the Laplace transform of their orthogonality measure. In this paper, we study analogs of the latter three classes of polynomials in infinite dimensions. We fix as an underlying space a (non-compact) Riemannian manifold $X$ and an intensity measure $σ$ on it. We consider a Jacobi field in the extended Fock space over $L^2(X;σ)$, whose field operator at a point $x\in X$ is of the form $\di_x^†+…

多索引离散变量正交多项式与精确可解的生灭过程

Multi-indexed Orthogonal Polynomials of a Discrete Variable and Exactly Solvable Birth and Death Processes

本文基于8种离散变量多索引正交多项式，推导出精确可解的连续时间生灭过程及其离散时间版本，涵盖多种正交多项式类型。