Clebsch-Gordan Series and the Clebsch-Gordan Coefficients of O(2, 1) and SU(1, 1)

The Clebsch-Gordan series of the O(2, 1) group and its covering group SU(1, 1) are derived for all cases except that of the supplemental series. The continuable Clebsch-Gordan coefficients (or equivalently, the Wigner coefficients) are explicitly expressed in terms of the generalized hypergeometric function 3F2. The spectra in the decomposition of the product of the two principal series are discussed. The applications to the UIR of O(2, 2) are also studied.