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

TL;DR

This paper explores multivariate classical orthogonal polynomials and introduces an infinite-dimensional extension, revealing their probabilistic relationships with other polynomial families through gamma point processes.

Contribution

It provides a novel infinite-dimensional version of Jacobi polynomials linked to gamma point processes and establishes probabilistic connections among multivariate orthogonal polynomials.

Findings

Hahn and Meixner polynomials as posterior mixtures of Jacobi and Laguerre polynomials

Construction of an infinite-dimensional Jacobi polynomial version using gamma point processes

Probabilistic interpretation of polynomial relationships

Abstract

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.