Realizations of coupled vectors in the tensor product of representations of and

Using the realization of positive discrete series representations of in terms of a complex variable z, we give an explicit expression for coupled basis vectors in the tensor product of ν+1 representations as polynomials in ν+1 variables z₁,…,z_ν+1. These expressions use the terminology of binary coupling trees (describing the coupled basis vectors), and are explicit in the sense that there is no reference to the Clebsch–Gordan coefficients of . In general, these polynomials can be written as (terminating) multiple hypergeometric series. For ν=2, these polynomials are triple hypergeometric series, and a relation between the two binary coupling trees yields a relation between two triple hypergeometric series. The case of is discussed next. Also here the polynomials are determined explicitly in terms of a known realization; they yield an efficient way of computing coupled basis vectors in terms of uncoupled basis vectors.