Wilson function transforms related to Racah coefficients

TL;DR

This paper computes Racah coefficients for $su(1,1)$ tensor products, expressing them as Wilson polynomials and functions, and explores their applications and q-deformations involving Askey-Wilson functions.

Contribution

It provides explicit formulas for Racah coefficients in terms of Wilson functions and extends the results to quantum groups with Askey-Wilson functions.

Findings

Racah coefficients are Wilson polynomials and functions

Derived new sum and integral identities involving Wilson functions

Computed Racah coefficients for quantum group $U_q(su(1,1))$ as Askey-Wilson functions

Abstract

The irreducible $*$-representations of the Lie algebra $su(1,1)$ consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the Clebsch-Gordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application several sum and integral identities are obtained involving Wilson polynomials and Wilson functions. We also compute Racah coefficients for $U_q(\su(1,1))$, which turn out to be Askey-Wilson functions and Askey-Wilson polynomials.