Multidimensional q-Normal and related distributions - Markov case

TL;DR

This paper introduces and analyzes multidimensional q-Normal distributions, exploring their properties, generalizations, and connections to Askey-Wilson polynomials, including a generalized Poisson-Mehler expansion formula.

Contribution

It defines multidimensional q-Normal distributions, studies their properties, and links them to Askey-Wilson weight functions, extending classical results.

Findings

q=1 case recovers multidimensional Normal distribution

Distributions have densities and compact support for q in (-1,1)

Generalized Poisson-Mehler expansion formula established

Abstract

We define and study distributions in R^{d} that we call q-Normal. For q=1 they are really multidimensional Normal, for q\in(-1,1) they have densities, compact support and many properties that resemble properties of ordinary multidimensional Normal distribution. We also consider some generalizations of these distributions and indicate close relationship of these distributions to Askey-Wilson weight function i.e. weight with respect which Askey-Wilson polynomials are orthogonal and prove some properties of this weight function. In particular we prove a generalization of Poisson-Mehler expansion formula