Characterizations of Orthogonal Polynomials Satisfying Differential Equations

Abstract

In 1938, H. L. Krall found a necessary and sufficient condition for an orthogonal polynomial set $\{ P_n (x)\} _0^\infty $ to satisfy a linear differential equation of the form \[ \sum_{i = 0}^N {\ell _i (x)y^{(i)} } (x) = \lambda _n y(x).\] Here the authors give a new simple proof of Krall’s theorem as well as some other characterizations of such orthogonal polynomial sets based on the symmetrizability of the differential operator. In particular it is shown that such orthogonal polynomial sets are characterized by a certain Sobolev-type orthogonality, which generalizes Hahn’s charaterization of classical orthogonal polynomials.