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

Abstract

We associate a family of Hilbert spaces Hq2;λ(D) of analytic functions on the unit disk D=z∈\(\mathbb{C}\):|z|<1 the q-continuous Gegenbauer polynomials Cnλ(x;q) on the interval]−1;1[ and give a q-analogue of the unitary integral transform that Watanabe constructed from the Hilbert space L2(]−1;1[;(1−x2)λ−\(\frac{1}{2}\)dx onto the weighted Hilbert space H2;λ(D).