Approach of the continuous q-Hermite polynomials to x-representation of q-oscillator algebra and its coherent states

We use the recursion relations of the continuous q-Hermite polynomials and obtain the q-difference realizations of the ladder operators of a q-oscillator algebra in terms of the Askey-Wilson operator. For q-deformed coherent states associated with a disc in the radius 1/root 1-q, we obtain a compact form in x-representation by using the generating function of the continuous q-Hermite polynomials, too. In this way, we obtain a q-difference realization for the q-oscillator algebra in the finite interval -root 3/1-q < x < root 2/1-q as a q-generalization of known differential formalism with respect to x in the interval -infinity < x < +infinity of the simple harmonic oscillator.