On the $D_{\omega}$-classical orthogonal polynomials

Abstract

We wish to investigate the $D_{\omega}$-classical orthogonal polynomials, where $D_{\omega}$ is a special case of the Hahn operator. For this purpose, we consider the problem of finding all sequences of orthogonal polynomials such that their $D_{\omega}$-derivatives are also orthogonal polynomials. To solve this problem we adopt a different approach to those employed in this topic. We first begin by determining the coefficients involved in their recurrence relations, and then providing an exhaustive list of all solutions. When $\omega=0$, we rediscover the classical orthogonal polynomials of Hermite, Laguerre, Bessel and Jacobi. For $\omega=1$, we encounter the families of discrete classical orthogonal polynomials as particular cases.