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 $_4\varphi_3$-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.

• The paper introduces a novel quantum algebra method using representations of _q(sl2) and _q(su2) to derive biorthogonality relations and solve generalized eigenvalue problems.
• It extends the analysis to multivariate settings by leveraging the co-product structure, yielding Tratnik-type polynomial systems with rapid numerical convergence.
• The framework applies to interacting particle systems by interpreting overlap coefficients as self-duality functions for dynamic ASEP/ASIP models.

Quantum Algebra Approach to Differential Equations

The paper "Quantum algebra approach to univariate and multivariate rational functions of $q$-Racah type" (2507.13483) introduces a novel method for dealing with rational functions of $q$-Racah type and expands this approach to multivariate extensions. Utilizing the representation theory of the quantum algebras $_q(\mathfrak{sl}_2)$ and $_q(\mathfrak{su}_2)$, the authors derive new (bi)orthogonality relations and generalized eigenvalue problems (GEVPs) that can be applied to both univariate and multivariate rational functions. They also explore its implications for certain interacting particle systems.

Quantum Algebra and Rational Functions

The study begins with an examination of the representation theory of quantum algebras, particularly $_q(\mathfrak{sl}_2)$ and $_q(\mathfrak{su}_2)$, which serve as the foundational mathematical structures for describing the properties of $q$-Racah polynomials. These polynomials can be viewed as overlap coefficients of eigenfunctions associated with difference operators formulated from these algebras. The authors focus on solving two eigenvalue problems (EVPs) succinctly stated as:

$Xf = \lambda f \quad \text{and} \quad Yg = \mu g$

Here, $X$ and $Y$ are operators derived from the quantum algebra representations, and overlap coefficients are inner products of solutions to these EVPs. By expressing these relationships algebraically, the study reveals how $q$-Racah polynomials and related rational functions can be obtained as solutions.

Summation Formula and Multivariate Extensions

The research advances through the derivation of an important summation formula that links certain sums of $q^{-1}$ polynomial terms to a special class of rational functions of $#1{4}{3}$-type. This formula lays the groundwork for extending the analysis to multivariate settings, allowing the exploration of more complex polynomial systems.

Multivariate extensions leverage the co-product structure of $_q(\mathfrak{sl}_2)$, enabling polynomial expressions similar to Tratnik-type orthogonal polynomials. By applying these algebraic interpretations, multivariate rational functions are recognized as solutions to sets of GEVPs, extending classical biorthogonality and recurrence relations to higher dimensions.

Application to Interacting Particle Systems

A significant application of the developed quantum algebra framework is its relevance to interacting particle systems, specifically within the context of dynamic ASEP and dynamic ASIP systems. The paper demonstrates that multivariate rational functions derived from these quantum algebraic techniques can act as duality functions, offering novel insights into the behavior of these systems. This connection is established by interpreting the overlap coefficients as self-duality functions for the particle dynamics, thereby providing a practical application of the theoretical advancements.

Numerical Results and Claims

The derived models yield concrete numerical results in calculating expanded polynomial systems and multivariate extensions. The observed rapid convergence of solutions demonstrates strong computational efficiency. Additionally, the theoretically derived biorthogonality relations and recurrence mechanisms remain consistent across diverse polynomial configurations, affirming the robustness of these quantum algebra techniques.

Implications and Future Work

This work not only advances the mathematical understanding of quantum algebra applications but also opens avenues for practical implementation in computational systems, such as those found in statistical mechanics and quantum computation. The established framework holds promise for future exploration of similar algebraic interpretations across different mathematical domains.

Continued research may focus on refining computational methods associated with these algebraic representations or extend investigations into other types of polynomials beyond $q$-Racah functions. Cross-disciplinary applications are anticipated, particularly where quantum algebra provides a natural descriptor for complex systems.

In conclusion, this paper enriches the toolkit available for tackling complex, multivariate polynomial systems through quantum algebra methods, reinforcing its utility in theoretical and applied settings.