A characterization of Dunkl-classical [formula omitted]-symmetric [formula omitted]-orthogonal polynomials and its applications

Abstract

In this paper, we obtain a necessary and sufficient condition on a linear operator J , defined on polynomials, and a d -symmetric d -orthogonal polynomial set { P n } n ≥ 0 such that { J P n } n ≥ 0 is also d -orthogonal. That allows us to characterize the Dunkl-classical d -symmetric d -orthogonal polynomials as the range of classical d -symmetric d -orthogonal polynomials by an operator related to the Dunkl operator. As applications, we derive many properties of the Dunkl-classical d -symmetric d -orthogonal polynomials from the classical d -symmetric d -orthogonal polynomials.