q-Algebra and q-Superalgebra Tensor Products and Identities for Special Function

Tensor products are constructed for distinct q generalizations of Euclidean oscillator- and sl(2)-type algebras and superalgebras, including cases where the method of highest weight vectors does not apply. In particular, three-term recurrence relations for Askey-Wilson polynomials are used to decompose the tensor product of representations from positive discrete series and representations from negative discrete series. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group, and the corresponding matrix elements of the group operators on these representation spaces are computed. The most important q-series identities derived here are interpreted as the expansion of the matrix elements of a group operator (via 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 and -q, respectively, for the algebra and superalgebra cases, where 0 < q < 1.