Casoratian Identities for the Wilson and Askey-Wilson Polynomials

TL;DR

This paper derives numerous Casoratian identities for Wilson and Askey-Wilson polynomials, establishing a foundation for eigenstate transformations in discrete quantum systems and extending to related polynomial families.

Contribution

It introduces a large set of Casoratian identities for Wilson and Askey-Wilson polynomials, linking eigenstate transformations to polynomial identities in quantum mechanics.

Findings

Derived infinitely many Casoratian identities for Wilson and Askey-Wilson polynomials.

Established the connection between these identities and Darboux transformations in quantum systems.

Extended identities to various reduced polynomial forms such as q-Jacobi and q-Hahn.

Abstract

Infinitely many Casoratian identities are derived for the Wilson and Askey-Wilson polynomials in parallel to the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials, which were reported recently by the present authors. These identities form the basis of the equivalence between eigenstate adding and deleting Darboux transformations for solvable (discrete) quantum mechanical systems. Similar identities hold for various reduced form polynomials of the Wilson and Askey-Wilson polynomials, e.g. the continuous q-Jacobi, continuous (dual) (q-)Hahn, Meixner-Pollaczek, Al-Salam-Chihara, continuous (big) q-Hermite, etc.