A generalization of classical symmetric orthogonal functions using a symmetric generalization of Sturm-Liouville problems

摘要

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