. C A ] 9 J ul 1 99 3 AN ANALOG OF THE FOURIER TRANSFORMATION FOR A q - HARMONIC OSCILLATOR

A q-version of the Fourier transformation and some of its properties are discussed. INTRODUCTION Models of q-harmonic oscillators are being developed in connection with quantum groups and their various applications (see, for example, Refs. [M], [Bi], [AS1], and [AS2]). For a complete correspondence with the quantum-mechanical oscillator problem, these models need an analog of the Fourier transformation that relates the coordinate and momentum spaces. In the present work we fill this gap for one of the models, the one based on the continuous q-Hermite polynomials [M], [AS1] when −1 < q < 1. In Section I we assemble all those formulas from [AS1], which are necessary for the subsequent exposition. In Section II we discuss the relation between the Mehler bilinear generating function for Hermite polynomials and the kernel exp ( i hpx ) of the Fourier transformation that connects the coordinate x and momentum p spaces [W]. We used the bilinear formula of L. J. Rogers to obtain a reproducing kernel and an analogue of the Fourier transform in the setting determined by the continuous q-Hermite polynomials of Rogers. Some properties of the q-Fourier transformation are discussed in Sections III-IV. In the following we take 0 < q < 1, although most of the formulas remain correct when −1 < q < 0. The limiting case q → 0 is of some mathematical interest. 1. q-Hermite functions. The continuous q-Hermite polynomials were introduced by Rogers [R]. They can be defined by the three term recurrence relation 2xHn(x | q) = Hn+1(x | q) + (1− q)Hn−1(x | q), (1.1) H0(x | q) = 1, H1(x | q) = 2x. They are orthogonal on −1 ≤ x = cos θ ≤ 1 with respect to a positive measure ρ(x) 1 ρ(x) = 4 sin θ(qe, qe; q)∞ (1.2) = 4 √