The Askey-Wilson polynomials and q-Sturm-Lioville problems

TL;DR

This paper explores the properties of Askey-Wilson polynomials by analyzing their associated q-Sturm-Liouville problems, providing new operator-theoretic insights and simplified derivations of their characteristics.

Contribution

It introduces the adjoint of the Askey-Wilson difference operator and characterizes the polynomials as solutions to a q-Sturm-Liouville problem, offering a new perspective.

Findings

Identified the adjoint of the Askey-Wilson operator.

Demonstrated that Askey-Wilson polynomials solve a q-Sturm-Liouville problem.

Provided an operator-theoretic framework for the Askey-Wilson operator.

Abstract

We find the adjoint of the Askey-Wilson divided difference operator with respect to the inner procuct on L^2(-1,1,(1-x^2)^-1/2 dx) defined as a Cauchy principle value and show that the Askey-Wilson polynomials are solutions of a q-Sturm-Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic description of the Askey-Wilson operator.