Quantum algebra approach to univariate and multivariate rational functions of $q$-Racah type

Abstract

In this paper, we study rational functions of $q$-Racah type and a multivariate extension, using representation theory of ${\mathcal{U}}_q({\mathfrak{sl}}_2)$. Eigenfunctions of twisted primitive elements in ${\mathcal{U}}_q({\mathfrak{su}}_2)$ can be expressed in terms of $q^{-1}$-Krawtchouk polynomials. Using this, we show that overlap coefficients of solutions of a generalized eigenvalue problem (GEVP) and an eigenvalue problem (EVP) can be expressed in terms of a rational function of ${}_4{\phi }_3$-type. With help of the quantum algebra, we derive (bi)orthogonality relations as well as a GEVP for these functions. Furthermore, using this new algebraic interpretation, we can exploit the co-algebra structure of ${\mathcal{U}}_q({\mathfrak{sl}}_2)$ to find a multivariate extension of these rational functions and derive biorthogonality relations and GEVPs for the multivariate functions. Then we repeat this procedure for the non-compact quantum algebra ${\mathcal{U}}_q({\mathfrak{su}}_{1,1})$, where the $q^{-1}$-Al-Salam--Chihara polynomials play the role of the $q^{-1}$-Krawtchouk polynomials. As an application of the multivariate rational functions, we show that they appear as duality functions for certain interacting particle systems.

27 pages