Wilson function transforms related to Racah coefficients
The irreducible \(*\)-representations of the Lie algebra su\((1,1)\) consist of discrete series representations, principal unitary series and complementary series. The author computes Racah coefficients for tensor products of representations where at least two of them belong to the discrete series. This is done by using explicit expressions for the Clebsch-Gordan coefficients in terms of hypergeometric functions. It turns out that the Racah coefficients are Wilson polynomials and functions which lead to a natural interpretation of the integral transforms with Wilson functions as kernels. As further applications, several identities on sums and integrals of Wilson functions are given. Moreover, it is shown that the same procedure for the quantizized universal enveloping algebra \(U_q(\text{su}(1,1))\) leads to Askey-Wilson functions and polynomials.
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