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arXiv · 2202.05925

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

Ismaël Bussière·Julien Gaboriaud·Luc Vinet·Alexei Zhedanov
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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 ${}_{10}ϕ_9$ biorthogonal rational functions are also discussed.

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Ismaël Bussière, Julien Gaboriaud, Luc Vinet, Alexei Zhedanov. 2022-05-30. Bispectrality and biorthogonality of the rational functions of $q$-Hahn type. https://doi.org/10.1016/j.jmaa.2022.126443

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