An eigenvalue problem for the associated Askey-Wilson polynomials

TL;DR

This paper develops a new eigenvalue problem for associated Askey-Wilson polynomials by constructing a two-variable auxiliary function and identifying operators that relate it to standard Askey-Wilson polynomials, using computer algebra.

Contribution

It introduces a novel eigenvalue problem for associated Askey-Wilson polynomials through the construction of a two-variable auxiliary function and operator analysis.

Findings

Derived an eigenvalue problem linking associated and standard Askey-Wilson polynomials.

Identified a new Askey-Wilson operator using computer algebra.

Established relationships between operators and polynomial families.

Abstract

To derive an eigenvalue problem for the associated Askey-Wilson polynomials, we consider an auxiliary function in two variables which is related to the associated Askey-Wilson polynomials introduced by Ismail and Rahman. The Askey-Wilson operator, applied in each variable separately, maps this function to the ordinary Askey-Wilson polynomials with different sets of parameters. A third Askey-Wilson operator is found with the help of a computer algebra program which links the two, and an eigenvalue problem is stated.