A q-analogue of the weighted Bergman space on the disk and associated second q-Bargmann transform

Abstract

We provide a q-analogue of the classical weighted Bergman space on the complex unit disk and we give the explicit expression of the corresponding reproducing kernel function. Moreover, we give a q-analogue of the second Bargmann integral transform introduced by V. Bargmann in [4, p. 203]. We show that it defines a unitary integral transform from the Hilbert space on the nonnegative real half line spanned by the q-Laguerre polynomials with respect to the weight function xαexpq⁡(−(1−q)x), onto the considered weighted q-Bergman Hilbert space.