The symmetric Dunkl-classical orthogonal polynomials revisited

Abstract

ABSTRACT We investigate the symmetric Dunkl-classical orthogonal polynomials by using a new approach in connection with the Dunkl operator. The main aim of this technique is first and foremost to determine the recurrence coefficients. We establish the existence and uniqueness of symmetric Dunkl-classical orthogonal polynomials, and confirm that the two families of generalized Hermite orthogonal polynomials and Gegenbauer orthogonal polynomials are the only ones belonging to this class. Two apparently new characterizations in a more general setting are given. This paper complements earlier work of Ben Cheikh and Gaied [Characterization of the Dunkl-classical symmetric orthogonal polynomials. Appl Math Comput. 2007;187:105–114. doi: 10.1016/j.amc.2006.08.108].