Algebras behind the bispectrality of the Wilson rational functions and their ${}_4\phi_3$ limits

TL;DR

This paper explores the algebraic structures underlying Wilson rational functions, their spectral properties, and their limits to simpler functions, revealing connections to known algebras like the meta q-Racah algebra.

Contribution

It introduces the Wilson rational algebra that encodes spectral properties and analyzes limits leading to ${}_4 ext{phi}_3$ functions, connecting to known algebraic structures.

Findings

Wilson rational functions satisfy specific recurrence and eigenvalue relations.

The Wilson rational algebra encodes spectral properties of these functions.

Limits of Wilson functions relate to ${}_4 ext{phi}_3$ functions and simplify to known algebras.

Abstract

The properties of the Wilson rational functions ${}_{10}\phi_9$ with three different normalizations are described. For one normalization, it satisfies an $R_{II}$ recurrence relation, whereas for the two other ones, they satisfy a generalized eigenvalue problem. The so-called Wilson rational algebra is introduced, which encodes algebraically the spectral properties of these special functions. Finally, different limits are considered, leading up to functions proportional to ${}_{4}\phi_3$. For one of these, the spectral algebra simplifies to yield the meta $q$-Racah algebra.