The Gel’fand realization and the generating function of the Clebsch–Gordan coefficients of SL(2,R) in the hyperbolic basis

Abstract

It is shown that the canonical realization of the representations of SL(2,R) proposed by Gel’fand and co-workers yields a generating function of the Clebsch–Gordan coefficients of the group in the hyperbolic basis. This function is the coupled state and appears as the solution of an ordinary differential equation reducible to the hypergeometric equation. The desired expansion of the generating function that yields the Clebsch–Gordan coefficients is essentially a generalization of Barnes’ theory of analytic continuation of the hypergeometric function. In this paper the normalized Clebsch–Gordan coefficients for the coupling of two representations of the positive discrete class are calculated. The final result is an analytic continuation of the corresponding expression in the SO(2) basis. The possible application of the generating function to the reduction of the Kronecker product of three irreducible representations is discussed.