On the families of polynomials forming a part of the Askey--Wilson scheme and their probabilistic applications

TL;DR

This paper reviews six families of orthogonal polynomials within the Askey--Wilson scheme, highlighting their properties, interrelations, and applications in probabilistic models to generalize Gaussian distributions.

Contribution

It consolidates scattered recent results on these polynomials and their probabilistic applications, providing a comprehensive overview and new insights into their roles in distribution generalizations.

Findings

Connection coefficients between polynomial families

Linearization formulas and identities

Probabilistic models involving these polynomials

Abstract

We review the properties of six families of orthogonal polynomials that form the main bulk of the collection called the Askey--Wilson scheme of polynomials. We give connection coefficients between them as well as the so-called linearization formulae and other useful important finite and infinite expansions and identities. An important part of the paper is the presentation of probabilistic models where most of these families of polynomials appear. These results were scattered within the literature in recent 15 years. We put them together to enable an overall outlook on these families and understand their crucial r\^{o}le in the attempts to generalize Gaussian distributions and find their bounded support generalizations. The paper is based on 65 items in the predominantly recent literature.