Transforms between multivariate Laguerre and Meixner polynomials and their applications to gateways

Abstract

We establish gateways in both directions between the Laguerre process and a discrete Markov process for every inverse temperature $2\theta \ge 1$. Here, a gateway is an intertwining relation between Markov processes with continuous and discrete state spaces. A key ingredient is a pair of generating function identities that relate two double generating functions for multivariate Meixner polynomials to a single and a double generating function for multivariate Laguerre polynomials, respectively. These identities reveal a close relationship between the two families of polynomials and yield integral transforms that underlie the gateway relations. Gateway relations provide a means of transferring properties between continuous and discrete processes. As an application, we derive the Feller-Dynkin property and the uniqueness of the invariant distribution for the Laguerre process from the corresponding properties of the discrete process.

arXiv comment40 pages, no figures