  We discuss a model of a $q$-harmonic oscillator based on Rogers-Szeg\"o
functions. We combine these functions with a class of $q$-analogs of complex
Hermite polynomials to construct a new set of coherent states depending on a
nonnegative integer parameter $m$. Our construction leads to a new
$q$-deformation of the $m$-true-polyanalytic Bargmann transform whose range
defines a generalization of the Arik-Coon space. We also give an explicit
formula for the reproducing kernel of this space. The obtained results may be
exploited to define a $q$-deformation of the Ginibre-$m$-type process on the
complex plane.
