New characterizations of classical orthogonal polynomials

Abstract

Classical orthogonal polynomials of Jacobi, Laguerre, Hermite, and Bessel are characterized as the only orthogonal polynomials (up to a linear change of variable) such that (i) (Bochner) they satisfy a second order differential equation of the form l(2)(x)y ''(x)+l(1)(x)y'(x) = lambda(n)y(x); and (ii) (Hahn) their derivatives of any fixed order are also orthogonal. Here, we give several new characterizations of classical orthogonal polynomials including extensions of the above two characterizations.