On the structure and probabilistic interpretation of Askey-Wilson densities and polynomials with complex parameters

TL;DR

This paper explores the structure and probabilistic interpretation of Askey-Wilson polynomials with complex parameters, providing new formulas and connections to free probability, and enabling explicit moment calculations.

Contribution

It introduces equivalent forms of Askey-Wilson polynomials using Al-Salam-Chihara polynomials and expands the Askey-Wilson weight function with a probabilistic interpretation.

Findings

Expanded Askey-Wilson weight function in a Poisson-Mehler-like series

Derived explicit formulas for q-Hermite moments of the Askey-Wilson density

Connected Askey-Wilson polynomials to free probability through parameter limits

Abstract

We give equivalent forms of the Askey-Wilson polynomials expressing them with the help of the Al-Salam-Chihara polynomials. After restricting parameters of the Askey-Wilson polynomials to complex conjugate pairs we expand the Askey-Wilson weight function in the series similar to the Poisson-Mehler expansion formula and give its probabilistic interpretation. In particular this result can be used to calculate explicit forms of 'q-Hermite' moments of the Askey-Wilson density, hence enabling calculation of all moments of the Askey-Wilson density. On the way (by setting certain parameter q to 0) we get some formulae useful in the rapidly developing so called 'free probability'.