Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials

TL;DR

This paper introduces infinitely many new exceptional orthogonal polynomials linked to Wilson and Askey-Wilson polynomials, derived from exactly solvable quantum systems with shape invariance, expanding the class of orthogonal polynomials beyond classical constraints.

Contribution

It constructs new families of exceptional orthogonal polynomials associated with Wilson and Askey-Wilson polynomials via shape invariant quantum Hamiltonians, not limited by Bochner's theorem.

Findings

Presented two sets of exceptional orthogonal polynomials.

Derived from shape invariant, exactly solvable quantum Hamiltonians.

Polynomials start from degree ≥1, not constrained by Bochner's theorem.

Abstract

Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey-Wilson polynomials in terms of a degree \ell (\ell=1,2,...) eigenpolynomial. These polynomials are exceptional in the sense that they start from degree \ell\ge1 and thus not constrained by any generalisation of Bochner's theorem.