Difference Schrödinger equation and $q$-oscillator model

A model of $q$-oscillator, wave functions of which in the relativistic configurational $x$-space are expressed through the $q$-Hermite–Szegö polynomials and in the momentum $p$-space – through the Stieltjes–Wigert polynomials is considered. Some properties of the $q$-Hermite–Szegö polynomials are studied.