Bispectrality and biorthogonality of the rational functions of $q$-Hahn type

TL;DR

This paper introduces biorthogonal rational functions related to the $q$-Hahn family, explores their bispectral properties via a triplet of $q$-difference operators, and connects them to Wilson's biorthogonal functions.

Contribution

It presents a new family of biorthogonal rational functions with bispectral properties and analyzes their algebraic and operator structures, extending the theory of $q$-orthogonal polynomials.

Findings

Defined biorthogonal rational functions with respect to the $q$-hypergeometric distribution.

Established a triplet of $q$-difference operators acting as bispectral operators.

Connected the new functions to Wilson's ${}_{10} ext{phi}_9$ biorthogonal rational functions.

Abstract

We introduce families of rational functions that are biorthogonal with respect to the $q$-hypergeometric distribution. A triplet of $q$-difference operators $X$, $Y$, $Z$ is shown to play a role analogous to the pair of bispectral operators of orthogonal polynomials. The recurrence relation and difference equation take the form of generalized eigenvalue problems involving the three operators. The algebra generated by $X$, $Y$, $Z$ is akin to the algebras of Askey--Wilson type in the case of orthogonal polynomials. The actions of these operators in three different basis are presented. Connections with Wilson's ${}_{10}\phi_9$ biorthogonal rational functions are also discussed.