A realization of the q-deformed harmonic oscillator: Rogers-Szego and Stieltjes-Wigert polynomials

We discuss some results from q-series that can account for the foundations for the introduction of orthogonal polynomials on the circle and on the line, namely the Rogers-Szego and Stieltjes-Wigert polynomials. These polynomials are explicitly written and their orthogonality is verified. Explicit realizations of the raising and lowering operators for these polynomials are introduced in analogy to those of the Hermite polynomials that are shown to obey the q-commutation relations associated with the q-deformed harmonic oscillator.