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

Multivariate versions of classical orthogonal polynomials such as Jacobi, Hahn, Laguerre, 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, an 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. 1 Introduction. The Dirichlet distribution Dα on d < ∞ points, where α = (α1,..., αd) ∈ Rd+, is the probability distribution on the (d − 1)−dimensional simplex ∆(d−1): = {(x1,..., xd−1) ∈ [0, 1]d−1: d−1