Abstract
Gasper & Rahman's multivariate q-Racah polynomials are shown to arise as connection coefficients between families of multivariate q-Hahn or q-Jacobi polynomials. The families of q-Hahn polynomials are constructed as nested Clebsch–Gordan coefficients for the positive-discrete series representations of the quantum algebra suq(1,1). This gives an interpretation of the multivariate q-Racah polynomials in terms of 3nj symbols. It is shown that the families of q-Hahn polynomials also arise in wavefunctions of q-deformed quantum Calogero–Gaudin superintegrable systems.
Highlights
This paper shows that Gasper & Rahman’s multivariate q-Racah polynomials arise as the connection coefficients between two families of multivariate q-Hahn or qJacobi polynomials
The Gasper–Rahman scheme of multivariate q-orthogonal polynomials should be distinguished from the other multivariate extension of the Askey scheme based on root systems, which includes the Macdonald–Koornwinder polynomials [25] and the q-Racah polynomials defined by van Diejen and Stokman [36]
We have shown that the Gasper–Rahman multivariate q-Racah polynomials arise as the connection coefficients between these bases of qHahn and q-Jacobi polynomials, and we have provided an interpretation for these polynomials in terms of special 3n j-coefficients for suq(1, 1)
Summary
This paper shows that Gasper & Rahman’s multivariate q-Racah polynomials arise as the connection coefficients between two families of multivariate q-Hahn or qJacobi polynomials. The two families of q-Hahn polynomials are constructed as nested Clebsch–Gordan coefficients for the positive-discrete series representations of the quantum algebra suq(1, 1). This result gives an algebraic interpretation of the multivariate q-Racah polynomials as recoupling coefficients, or 3n j-symbols, of suq(1, 1). Upon considering the 3-fold tensor product representations of su(1, 1), one finds that the two intermediate Casimir operators associated to adjacent pairs of representations in the tensor product satisfy the (rank one) Racah algebra, which is the algebra generated by the two operators involved in the bispectral property of the univariate Racah polynomials. The bases will be constructed using the nested Clebsch–Gordan coefficients for multifold tensor product representations of suq(1, 1), which will provide the exact interpretation of the multivariate q-Racah polynomials in terms of coupling coefficients for that quantum algebra. For a given value of d, the Casimir operator Γ[1;d] will be referred to as the full Casimir operator
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