A basic class of symmetric orthogonal functions using the extended Sturm–Liouville theorem for symmetric functions

Abstract

By using the extended Sturm–Liouville theorem for symmetric functions, we introduced a basic class of symmetric orthogonal polynomials (BCSOP) in a previous paper. The mentioned class satisfies a differential equation of the form x 2 ( px 2 + q ) Φ n ″ ( x ) + x ( rx 2 + s ) Φ n ′ ( x ) - ( n ( r + ( n - 1 ) p ) x 2 + ( 1 - ( - 1 ) n ) s / 2 ) Φ n ( x ) = 0 and contains four main sequences of symmetric orthogonal polynomials. In this paper, again by using the mentioned theorem, we introduce a basic class of symmetric orthogonal functions (BCSOF) as a generalization of BCSOP and obtain its standard properties. We show that the latter class satisfies the equation x 2 ( px 2 + q ) Φ n ″ ( x ) + x ( rx 2 + s ) Φ n ′ ( x ) - ( α n x 2 + ( 1 - ( - 1 ) n ) β / 2 ) Φ n ( x ) = 0 , in which β is a free parameter and - α n denotes eigenvalues corresponding to BCSOF. We then consider four sub-classes of defined orthogonal functions class and study their properties in detail. Since BCSOF is a generalization of BCSOP for β = s , the four mentioned sub-classes respectively generalize the generalized ultraspherical polynomials, generalized Hermite polynomials and two other finite sequences of symmetric polynomials, which were introduced in the previous work.