Sturm-Liouville Theory and Orthogonal Functions

TL;DR

This paper revisits classical Sturm-Liouville theory, providing new insights into polynomial solutions, weight functions, and classifications of differential equations, including higher-order cases and explicit examples.

Contribution

It offers a new argument for Bochner's classification and extends the analysis to higher-order equations with polynomial solutions.

Findings

Recovered Bochner's classification using a new approach

Extended the theory to higher-order differential equations

Provided explicit examples of equations with polynomial solutions

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.