q-Analogue of the Watanabe Unitary Transform Associated to the q-Continuous Gegenbauer Polynomials

Abstract

We associate a family of Hilbert spaces H _q ^2;λ(D) of analytic functions on the unit disk D=z∈$\mathbb{C}$:|z|<1 the q-continuous Gegenbauer polynomials C _n ^λ(x;q) on the interval]−1;1[ and give a q-analogue of the unitary integral transform that Watanabe constructed from the Hilbert space L ²(]−1;1[;(1−x ²)^λ− $\frac{1}{2}$dx onto the weighted Hilbert space H ^2;λ(D).