MODELS OF Q-ALGEBRA REPRESENTATIONS - MATRIX-ELEMENTS OF THE Q-OSCILLATOR ALGEBRA

This article continues a study of function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. Here a q analog of the oscillator algebra (not a quantum algebra) is consider ed. 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 Li e group and the corresponding matrix elements of the ''group operators '' on these representation spaces are computed. This ''local'' approac h applies to more general families of special functions, e.g., with co mplex arguments and parameters, than does the quantum group approach. It is shown that the matrix elements themselves transform irreducibly under the action of the algebra. q analogs of a formula are found for the product of two hypergeometric functions F-1(1) and the product of a F-1(1) and a Bessel function. They are interpreted here as expansion s of the matrix elements of a ''group operator'' (via the exponential mapping) in a tensor product basis (for the tensor product of two irre ducible oscillator algebra representations) in terms of the matrix ele ments in a reduced basis. As a by-product of this analysis an interest ing new orthonormal basis was found for a q analog of the Bargmann-Seg al Hilbert space of entire functions.