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

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 ${}_{10}{\varphi }_9$ biorthogonal rational functions are also discussed.

arXiv comment19 pages, updated version to match the one accepted for publication

References in corpus (5)

• The q-harmonic oscillator and the Al-Salam and Carlitz polynomials
• Bispectral commuting difference operators for multivariable Askey-Wilson polynomials
• Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions
• Degenerate Sklyanin algebras, Askey-Wilson polynomials and Heun operators
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials

Cited by in corpus (2)

• Meta Algebras and Biorthogonal Rational Functions: The Hahn Case
• $\lambda$-Griffiths polynomials: Bispectrality and biorthogonality