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(alpha,beta)-classical orthogonal polynomials, where J(alpha,beta) is the raising operator J(alpha,beta) := (x(2) - 1) d/dx + (alpha + beta) x - alpha + beta)I, with alpha and beta nonzero complex numbers and I representing the identity operator. Then, we show that the Jacobi polynomials P-n((alpha,beta)) (x), n >= 0, where alpha, beta is an element of C\{0,-1,-2,...}, (alpha + beta not equal - m, m >= 0), are the only J(alpha,beta)-classical orthogonal polynomials. As an application, we give some new connection formulas satisfied by the polynomials solution of our problem.