BESSEL POLYNOMIALS AND SOME CONNECTION FORMULAS IN TERMS OF THE ACTION OF LINEAR DIFFERENTIAL OPERATORS - Baghdadi Aloui; Jihad Souissi (Ural Mathematical Journal)
In this paper, we introduce the concept of the \(\mathbb{B}_{α}\)-classical orthogonal polynomials, where \(\mathbb{B}_{α}\) is the raising operator \(\mathbb{B}_{α}:=x^2 · {d}/{dx}+\big(2(α-1)x+1\big)\mathbb{I}\), with nonzero complex number \(α\) and \(\mathbb{I}\) representing the identity operator. We show that the Bessel polynomials \(B^{(α)}_n(x),\ n≥0\), where \(α≠-{m}/{2}, \ m≥ -2, \ m\in \mathbb{Z}\), are the only \(\mathbb{B}_{α}\)-classical orthogonal polynomials. As an application, we present some new formulas for polynomial solution.