Tensor products of q-superalgebra representations and q-series identities

We work out examples of tensor products for distinct q-generalizations of Euclidean, oscillator and sℓ(2) type superalgebras in cases where the method of highest-weight vectors will not apply. In particular, we use the three-term recurrence relations for Askey-Wilson polynomials to decompose the tensor product of representations from the positive discrete series and representations from the negative discrete series. We show that various q-analogues of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and we compute the corresponding matrix elements of the 'group operators' on these representation spaces. We show that the matrix elements themselves transform irreducibly under the action of the quantum superalgebra. The most important q-series identities derived here are interpreted as the expansion of the matrix elements of a 'group operator' (via the exponential mapping) in a tensor product basis in terms of the matrix elements in a reduced basis They involve q-hypergeometric series with base -q, 0 < q < 1.