A model for the continuous q-ultraspherical polynomials
Abstract
We provide an algebraic interpretation for two classes of continuous q-polynomials. Rogers' continuous q-Hermite polynomials and continuous q-ultraspherical polynomials are shown to realize, respectively, bases for representation spaces of the q-Heisenberg algebra and a q-deformation of the Euclidean algebra in these dimensions. A generating function for the continuous q-Hermite polynomials and a q-analog of the Fourier-Gegenbauer expansion are naturally obtained from these models.
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