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

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, Zis akin to the algebras of Askey-Wilson type in the case of orthogonal polynomials. The actions of these operators in three different bases are presented. Connections with Wilson's (10)phi(9) biorthogonal rational functions are also discussed. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this paper, a family of rational functions biorthogonal to q-hypergeometric distribution is introduced, showing that a triplet of q-difference operators X, Y, Z plays a role analogous to bispectral operators of orthogonal polynomials. The recurrence relation and difference equation are presented as generalized eigenvalue problems involving the three operators, and the algebra generated by X, Y, Z is similar to Askey-Wilson type algebras in the case of orthogonal polynomials. The actions of these operators in three bases and connections with Wilson's (10)phi(9) biorthogonal rational functions are discussed.