Clebsch-Gordan coefficients and special function identities. I. The harmonic oscillator group

It is shown that by constructing explicit realizations of the Clebsch-Gordan decomposition for tensor products of irreducible representations of a group G, one can derive a wide variety of special function identities with physical interest. In this paper, the representation theory of the harmonic oscillator group is used to give elegant derivations of identities involving Hermite, Laguerre, Bessel, and hypergeometric functions.