A model for the continuous q-ultraspherical polynomials

Details A model for the continuous q-ultraspherical polynomials A model for the continuous q-ultraspherical polynomials

1995-04-26

arxiv.org/abs/math/9504219v1

Mathematics

Classical Analysis and ODEs

Scientific paper

10.1063/1.530998

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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