Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$

TL;DR

This paper explores the relationship between Racah polynomials, $rak{su}(1,1)$ recoupling schemes, and superintegrable systems, revealing algebraic structures and bispectral properties of these polynomials.

Contribution

It establishes a connection between Racah polynomials and $rak{su}(1,1)$ recoupling schemes, extending the quadratic algebra and analyzing bispectrality.

Findings

Racah polynomials serve as connection coefficients in different coordinate systems.

An extended quadratic algebra ${ m QR}(3)$ is introduced and closed with an additional shift operator.

The duality and bispectrality of the polynomials are interpreted through expansion coefficients.

Abstract

The connection between the recoupling scheme of four copies of $\mathfrak{su}(1,1)$, the generic superintegrable system on the 3 sphere, and bivariate Racah polynomials is identified. The Racah polynomials are presented as connection coefficients between eigenfunctions separated in different spherical coordinate systems and equivalently as different irreducible decompositions of the tensor product representations. As a consequence of the model, an extension of the quadratic algebra ${\rm QR}(3)$ is given. It is shown that this algebra closes only with the inclusion of an additional shift operator, beyond the eigenvalue operators for the bivariate Racah polynomials, whose polynomial eigenfunctions are determined. The duality between the variables and the degrees, and hence the bispectrality of the polynomials, is interpreted in terms of expansion coefficients of the separated solutions.