Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations

TL;DR

This paper presents a new derivation of exceptional Askey-Wilson and Wilson polynomials using Darboux-Crum transformations, highlighting their shape invariance and extending to continuous Hahn polynomials.

Contribution

It introduces a simplified method for deriving exceptional orthogonal polynomials and demonstrates their shape invariance, expanding the class of known exceptional polynomials.

Findings

Derivation of exceptional Wilson and Askey-Wilson polynomials

Extension to continuous Hahn polynomials

Proof of shape invariance for these systems

Abstract

An alternative derivation is presented of the infinitely many exceptional Wilson and Askey-Wilson polynomials, which were introduced by the present authors in 2009. Darboux-Crum transformations intertwining the discrete quantum mechanical systems of the original and the exceptional polynomials play an important role. Infinitely many continuous Hahn polynomials are derived in the same manner. The present method provides a simple proof of the shape invariance of these systems as in the corresponding cases of the exceptional Laguerre and Jacobi polynomials.