Bessel polynomials and some connection formulas in terms of the action of linear differential operators

In this paper, we introduce the concept of the $\mathbb{B}_{\alpha}$-classical orthogonal polynomials, where $\mathbb{B}_{\alpha}$ is the raising operator $\mathbb{B}_{\alpha}:=x^2 \cdot {d}/{dx}+\big(2(\alpha-1)x+1\big)\mathbb{I}$, with nonzero complex number $\alpha$ and $\mathbb{I}$ representing the identity operator. We show that the Bessel polynomials $B^{(\alpha)}_n(x),\ n\geq0$, where $\alpha\neq-{m}/{2}, \ m\geq -2, \ m\in \mathbb{Z}$, are the only $\mathbb{B}_{\alpha}$-classical orthogonal polynomials. As an application, we present some new formulas for polynomial solution.