A Generalization of Classical Symmetric Orthogonal Functions Using a Symmetric Generalization of Sturm-Liouville Problems

TL;DR

This paper extends Sturm-Liouville problems to symmetric functions, introducing generalized orthogonal solutions including a symmetric version of associated Legendre functions with broad applications.

Contribution

It proposes a symmetric generalization of Sturm-Liouville problems, leading to new orthogonal functions and differential equations with potential applications in physics.

Findings

Introduces a symmetric Sturm-Liouville framework

Derives generalized orthogonal functions including symmetric Legendre functions

Provides examples with broad applicability in physics

Abstract

In this paper, usual Sturm-Liouville problems are extended for symmetric functions so that the corresponding solutions preserve the orthogonality property. Two basic examples, which are special cases of a generalized Sturm-Liouville problem, are then introduced. First example generalizes the associated Legendre functions having extensive applications in physics and the second example introduces a generic differential equation with various sub-cases that have orthogonal solutions. For instance, this generic equation possesses a symmetric differential equation containing a basic solution of symmetric orthogonal polynomials.