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A generalization of classical symmetric orthogonal functions using a symmetric generalization of Sturm-Liouville problems. (English) Zbl 1133.34020

Summary: 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.

MSC:

34B24 Sturm-Liouville theory
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
33C47 Other special orthogonal polynomials and functions
34C14 Symmetries, invariants of ordinary differential equations
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