Characterization of polynomials via a raising operator

This paper investigates a first-order linear differential operator $\mathcal{J}_\xi$, where $\xi=(\xi_1, \xi_2) \in \mathbb{C}^2\setminus{(0, 0)}$, and $D:=\frac{d}{dx}$. The operator is defined as $\mathcal{J}_{\xi}:=x(xD+\mathbb{I})+\xi_1\mathbb{I}+\xi_2 D$, with $\mathbb{I}$ representing the identity on the space of polynomials with complex coefficients. The focus is on exploring the $\mathcal{J}_\xi$-classical orthogonal polynomials and analyzing properties of the resulting sequences. This work contributes to the understanding of these polynomials and their characteristics.