. C A ] 9 J ul 1 99 3 The q - Harmonic Oscillator and an Analog of the Charlier polynomials

A model of a q-harmonic oscillator based on q-Charlier poly-nomials of Al-Salam and Carlitz is discussed. Simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an ana-log of the Fourier transformation are found. A connection of the kernel of this transform with biorthogonal rational functions is observed. Models of q-harmonic oscillators are being developed in connection with quantum groups and their various applications (see, for example, Refs. [1–5]). The q-analogs of boson operators were introduced explicitly in Refs. [1,3] and [5], where the corresponding wave functions were found in terms of the continuous q-Hermite polynomials of Rogers [6,7] and in terms of the Stieltjes–Wigert polynomials [8,9], respectively. Here we introduce one more explicit realization of q-creation and q-annihilation operators with the aid of q-Charlier polynomials of Al-Salam and Carlitz [10]. The q-orthogonal polynomials V a n studied by Al-Salam and Carlitz may be considered as a q-version of the Charlier polynomials c µ n (s) (see, for example, [11,12]). To emphasize this analogy we use the notation c