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

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.