BESSEL POLYNOMIALS AND SOME CONNECTION FORMULAS IN TERMS OF THE ACTION OF LINEAR DIFFERENTIAL OPERATORS

In this paper, we introduce the concept of \(\mathbb{B}_{\alpha}\)-classical orthogonal polynomials, where \(\mathbb{B}_{\alpha}\) is 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 identity operator. We show that Bessel polynomials \(B^{(\alpha)}_n(x),\ n\geq0\), \(\alpha\neq-{m}/{2}, \ m\geq -2, m\in \mathbb{Z}\), are only polynomials. As an application, present some new formulas for polynomial solution.