Dunkl-supersymmetric orthogonal functions associated with classical orthogonal polynomials

We consider the eigenvalue problem associated with the Dunkl-type differential operator (in which the reflection operator R is involved) L = dx R + v(x), (v(-x) = -v(x)), in the context of supersymmetric quantum mechanical models. By solving this eigenvalue prob- lem with the help of known exactly solvable potentials, we construct several classes of func- tions satisfying certain orthogonality relations. We call them the Dunkl-supersymmetric (Dunkl- SUSY) orthogonal functions. These functions can be expressed in terms of the classical orthog- onal polynomials (COPs). The key feature of these functions is that they appear by pairs, i.e., Qn(x) and Qn(−x) are both the eigenfunctions of L. A general formulation of the Dunkl-SUSY orthogonal polynomials is also presented.