Q-ALGEBRA AND Q-SUPERALGEBRA TENSOR-PRODUCTS AND IDENTITIES FOR SPECIAL FUNCTION

Tensor products are constructed for distinct q generalizations of Eucl idean oscillator- and sl(2)-type algebras and superalgebras, including cases where the method of highest weight vectors does not apply. In p articular, three-term recurrence relations for Askey-Wilson polynomial s are used to decompose the tensor product of representations from pos itive discrete series and representations from negative discrete serie s. 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 o n these representation spaces are computed. The most important q-serie s identities derived here are interpreted as the expansion of the matr ix elements of a group operator (via exponential mapping) in a tenser- 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.