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

In this paper, we introduce the concept of the J_α,β-classical orthogonal polynomials, where J_α,β is the raising operator J_α,β := (x² − 1) d/dx + ((α + β)x − α + β)I, with α and β nonzero complex numbers and I representing the identity operator. Then, we show that the Jacobi polynomials P_n^(α,β)(x), n ≥ 0, where α, β ∈ C\{0, −1, −2,...}, (α + β 6= −m, m ≥ 0), are the only J_α,β-classical orthogonal polynomials. As an application, we give some new connection formulas satisfied by the polynomials solution of our problem.