Abstract
In this work, we give a new proof for the orthogonality of the Dunkl-Hermite polynomials [image omitted], when the index -1/2, via quasi-monomiality techniques and show that when-m-1/2-m+1/2, where m is a positive integer, [image omitted] is an orthogonal set with respect to a sesquilinear form which leads to a Krein space structure. Then, given a family of polynomials orthogonal with respect to a linear functional expressed in terms of iteratives of the Dunkl operator on the real line, we treat the problem of extending to a space of test functions which includes polynomials. As an example, we treat the extension of the weight function with respect to which the Dunkl-Hermite polynomials are orthogonal.