Differential equations and Sobolev orthogonality

Abstract

Consider (Sobolev) orthogonal polynomials which are orthogonal relative to a Sobolev bilinear form integral(R) p(x)q(x)d mu(x) + integral(R) p'(x)q'd nu(x), where d mu(x) and d nu(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.