On Sobolev bilinear forms and polynomial solutions of second-order differential equations

Given a linear second-order differential operator L equivalent to phi D-2 + psi D with non zero polynomial coefficients of degree atmost 2, a sequence of real numbers lambda(n), n >= 0, and a Sobolev bilinear form B(p, q) = Sigma(N)(k=0) < u(k), p((k)) q((k))>, N >= 0, where u(k), 0 <= k <= N, are linear functionals defined on polynomials, we study the orthogonality of the polynomial solutions of the differential equation L[y] = lambda(n) y with respect to B. We show that such polynomials are orthogonal with respect to B if the Pearson equations D(phi u(k)) = (.psi+ k phi') u(k), 0 <= k <= N, are satisfied by the linear functionals in the bilinear form. Moreover, we use our results as a general method to deduce the Sobolev orthogonality for polynomial solutions of differential equations associated with classical orthogonal polynomials with negative integer parameters.