On classical orthogonal polynomials and differential operators

It is well known that the four families of classical orthogonal polynomials (Jacobi, Bessel, Hermite and Laguerre) each satisfy an equation FPn(x) = lambda P-n(n)(x), n >= 0, for an appropriate second-order differential operator F. In this paper it is shown that any linear differential operator U which has the Jacobi, Bessel, Hermite or Laguerre polynomials as eigenfunctions has to be a polynomial with constant coefficients in the classical second-order operator F.