# Dunkl-Supersymmetric Orthogonal functions associated with classical orthogonal polynomials

## TL;DR

This paper introduces Dunkl-supersymmetric orthogonal functions derived from classical orthogonal polynomials, solving a Dunkl-type eigenvalue problem within supersymmetric quantum models, revealing paired eigenfunctions and a general polynomial framework.

## Contribution

It constructs Dunkl-supersymmetric orthogonal functions expressed via classical polynomials, highlighting their paired nature and providing a general formulation for these functions.

## Key findings

- Functions expressed in terms of classical orthogonal polynomials.
- Eigenfunctions appear in pairs, Qn(x) and Qn(-x).
- Provides a general framework for Dunkl-supersymmetric orthogonal polynomials.

## Abstract

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 problem with the help of known exactly solvable potentials, we construct several classes of functions satisfying certain orthogonality relations. We call them the Dunkl-supersymmetric (Dunkl-SUSY) orthogonal functions. These functions can be expressed in terms of the classical orthogonal 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.