On a q-extension of the Hermite polynomials H-n(x) with the continuous orthogonality property on R

Abstract

We study a polynomial sequence of q-extensions of the classical Hermite polynomials H-n(x), which satisfies continuous orthogonality on the whole real line R with respect to the positive weight function. This sequence can be expressed either in terms of the q-Laguerre polynomials L-n((alpha))(x

q), alpha = +/-1/2, or through the discrete q-Hermite polynomials (h) over tilde (n)(x

q) of type II.