An algebraic description of the bispectrality of the biorthogonal rational functions of Hahn type

TL;DR

This paper provides an algebraic framework for understanding the bispectrality of biorthogonal rational functions of Hahn type, revealing their connection to quadratic algebras similar to Askey-Wilson algebras.

Contribution

It introduces an algebraic description of bispectrality for these rational functions, linking difference operators to quadratic algebras akin to those in hypergeometric polynomial theory.

Findings

Operators X, Y, Z are tridiagonal in three bases

Pairwise commutators generate a quadratic algebra

Algebraic structure parallels Askey-Wilson type algebras

Abstract

The biorthogonal rational functions of the ${_3}F_2$ type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials. These properties are described by three difference operators $X,Y,Z$ which are tridiagonal with respect to three distinct bases of the relevant finite-dimensional space. The pairwise commutators of the operators $X,Y,Z$ generate a quadratic algebra which is akin to the algebras of Askey-Wilson type attached to hypergeometric polynomials.