h-Transforms and Orthogonal Polynomials

Random walks and orthogonal polynomials: some challenges

The study of several naturally arising "nearest neighbours" random walks benefits from the study of the associated orthogonal polynomials and their orthogonality measure. I consider extensions of this approach to a larger class of random walks. This raises a number of open problems.

Ultraspherical type generating functions for orthogonal polynomials

We characterize, up to a conjecture, probability distributions of all order finite moments having ultraspherical type generating functions for orthogonal polynomials.

Projection formulas for orthogonal polynomials

We prove a projection formula for the four-parameter family of orthogonal polynomials that are a reparameterization of the polynomials in the Askey-Wilson class. By carefully analyzing the recurrence relations we manage to avoid using the explicit expression for the orthogonality measure, which would be cumbersome due to the complexity of the reparameterization.

Quasi-birth-and-death processes and multivariate orthogonal polynomials

The aim of this paper is to study some models of quasi-birth-and-death (QBD) processes arising from the theory of bivariate orthogonal polynomials. First we will see how to perform the spectral analysis in the general setting as well as to obtain results about recurrence and the invariant measure of these processes in terms of the spectral measure supported on some domain $Ω\subset\mathbb{R}^d$. Afterwards, we will apply our results to several examples of bivariate orthogonal polynomials, namely product orthogonal polynomials, orthogonal polynomials on a parabolic domain and orthogonal polynom…

平面正交多项式作为第一类多重正交多项式

Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials

本文证明关于修正高斯测度的平面正交多项式在整数指数条件下也是围道上的第一类多重正交多项式，并由此导出若干等价的 Riemann-Hilbert 问题，同时用新技术建立了 Lee-Yang 基本恒等式。