Sturm-Liouville Theory and Orthogonal Functions

Abstract

We revisit basics of classical Sturm-Liouville theory and, as an application, recover Bochner's classification of second order ODEs with polynomial coefficients and polynomial solutions by a new argument. We also outline how a wider class of equations with polynomial solutions can be obtained by allowing the weight to become infinite at isolated points:the Jacobi equation, in general, is of this type. For higher order equations, we also give the basic analysis required for determining the weight functions and constraints on the coefficients which make the differential operator defined by the equation self adjoint for even orders and anti self adjoint in odd orders. We also give explicit examples of such equations.