A model for the continuous q-ultraspherical polynomials
arXiv:math/9504219 · doi:10.1063/1.530998
Abstract
We provide an algebraic interpretation for two classes of continuous -polynomials. Rogers' continuous -Hermite polynomials and continuous -ultraspherical polynomials are shown to realize, respectively, bases for representation spaces of the -Heisenberg algebra and a -deformation of the Euclidean algebra in these dimensions. A generating function for the continuous -Hermite polynomials and a -analog of the Fourier-Gegenbauer expansion are naturally obtained from these models.