Bessel Polynomials and Some Connection Formulas in Terms of the Action of Linear Differential Operators

Abstract

In this paper, we introduce the concept of the Bα-classical orthogonal polynomials, where Bα is the raising operator Bα := x2⋅d∕dx+2(α-1)x+1I, with nonzero complex number α and I representing the identity operator. We show that the Bessel polynomials Bn(α)(x), n ≥ 0, where α≠ - m∕2, m ≥-2, m ∈ ℤ, are the only Bα-classical orthogonal polynomials. As an application, we present some new formulas for polynomial solution.