Multi-indexed Wilson and Askey-Wilson Polynomials

TL;DR

This paper introduces multi-indexed Wilson and Askey-Wilson polynomials within discrete quantum mechanics, constructed via Darboux transformations and virtual state deletions, extending the class of multi-indexed orthogonal polynomials.

Contribution

It presents the first construction of multi-indexed Wilson and Askey-Wilson polynomials using discrete Darboux transformations in one-dimensional quantum mechanics.

Findings

Derived explicit forms of multi-indexed Wilson and Askey-Wilson polynomials.

Extended the framework of multi-indexed orthogonal polynomials to new classical families.

Connected the construction to virtual state solutions and Darboux transformations.

Abstract

As the third stage of the project multi-indexed orthogonal polynomials, we present, in the framework of 'discrete quantum mechanics' with pure imaginary shifts in one dimension, the multi-indexed Wilson and Askey-Wilson polynomials. They are obtained from the original Wilson and Askey-Wilson polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of 'virtual state solutions' of type I and II, in a similar way to the multi-indexed Laguerre, Jacobi and (q-)Racah polynomials reported earlier.