Abstract
Consider (Sobolev) orthogonal polynomials which are orthogonal relative to a Sobolev bilinear form ∫ R p(x)1(x)dμ(x)+ ∫ R p′(x)q′(x)dν(x) , where d μ( x) and d νv( 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.
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