Askey--Wilson Integral and its Generalizations

TL;DR

This paper expands the Askey--Wilson density using $q$-Hermite polynomials, deriving integral values and moments, and introduces a method to generate generalized densities with additional parameters.

Contribution

It presents a novel series expansion of the Askey--Wilson density and a systematic approach to generalize it with more parameters using recurrence relations.

Findings

Derived the value of the Askey--Wilson integral.

Computed $q$-Hermite moments of the AW density.

Established a framework for generating generalized AW densities.

Abstract

We expand the Askey--Wilson (AW) density in a series of products of continuous $q-$Hermite polynomials times the density that makes these polynomials orthogonal. As a by-product we obtain the value of the AW integral as well as the values of integrals of $q-$Hermite polynomial times the AW density ($q-$Hermite moments of AW density). Our approach uses nice, old formulae of Carlitz and is general enough to venture a generalization. We prove that it is possible and pave the way how to do it. As a result we obtain system of recurrences that if solved successfully gives a sequence of generalized AW densities with more and more parameters.