Multi-indexed Wilson and Askey-Wilson Polynomials

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.

💡 Research Summary

In this paper the authors extend the theory of multi‑indexed orthogonal polynomials to the Wilson and Askey‑Wilson families within the framework of discrete quantum mechanics (dQM) with pure‑imaginary shifts. The Wilson and Askey‑Wilson polynomials are the most general members of the (continuous) hypergeometric and q‑hypergeometric families, respectively, and they satisfy second‑order difference equations that can be interpreted as Schrödinger‑type equations in dQM. The central construction proceeds by introducing “virtual state solutions” – non‑normalizable eigenfunctions of the original difference operator – of two types (type I and type II). Type I virtual states are obtained by shifting the set of parameters downward, while type II are obtained by shifting them upward.

A finite index set 𝔇 = {d₁,…,d_M} is chosen, and for each d_j a virtual state of the prescribed type is deleted using the discrete analogue of the Darboux transformation, equivalently the Crum‑Krein‑Adler (CKA) scheme. The successive deletions generate a new family of Hamiltonians that are isospectral to the original one except for the removed virtual levels. The eigenfunctions of the deformed Hamiltonians are expressed as Casoratian determinants built from the M deleted virtual states and the original physical eigenfunction φ_n(x):

P_{𝔇,n}(x)=Cas