A characterization of Dunkl-classical dd-symmetric dd-orthogonal polynomials and its applications

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