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

TL;DR

This paper develops a quantum algebra framework to analyze rational functions of $q$-Racah type and their multivariate extensions, revealing new orthogonality relations, eigenvalue problems, and applications to interacting particle systems.

Contribution

It introduces a novel algebraic approach using $
_q( ext{sl}_2)$ to derive properties of rational functions of $q$-Racah type and extends these to multivariate cases with applications.

Findings

Derived biorthogonality relations for $q$-Racah type functions

Established eigenvalue problems and their solutions using quantum algebra

Connected multivariate rational functions to duality functions in particle systems

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 $_4\varphi_3$-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.