### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 057, 17 pages      arXiv:1504.03705      https://doi.org/10.3842/SIGMA.2015.057
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

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

Sarah Post
Department of Mathematics, University of Hawai`i at Mānoa, Honolulu, HI, 96822, USA

Received April 16, 2015, in final form July 14, 2015; Published online July 23, 2015

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.

Key words: orthogonal polynomials; Lie algebras; representation theory.

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