Differential equations and Sobolev orthogonality

Consider (Sobolev) orthogonal polynomials which are orthogonal relative to a Sobolev bilinear form ∫Rp(x)1(x)dμ(x)+∫Rp′(x)q′(x)dν(x), where dμ(x) and dνv(x) are signed Borel measures with finite moments. We give necessary and sufficient conditions under which such orthogonal polynomials satisfy a linear spectral differential equation with polynomial coefficients. We then find a sufficient condition under which such a differential equation is symmetrizable. These results can be applied to Sobolev-Laguerre polynomials found by Koekoek and Meijer.