Sobolev orthogonal polynomials and second-order differential equations

There has been a considerable amount of recent research on the subject of Sobolev orthogonal polynomials. In this paper, we consider the problem of when a sequence of polynomials that are orthogonal with respect to the (Sobolev) symmetric bilinear form (p,q)1 = integral(R) pqd mu(0) + integral(R) p'q'd mu(1) satisfies a second-order differential equation of the form a(2) (x)y"(x) + a(1)(x)y'(x)= lambda ny(x). We shall obtain necessary and sufficient conditions for this to occur. Moreover, we will characterize all sequences of polynomials satisfying these conditions. Included in this classification are some, in a sense, new orthogonal polynomials. As a consequence of this work, we obtain a new characterization of the classical orthogonal polynomials of Jacobi, Laguerre, Hermite, and Bessel.