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

• Authors: Mohammad Masjed-Jamei
• Venue: Journal of Computational and Applied Mathematics
• Year: 2008

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 formx^2(px^2+q)@F"n^''(x)+x(rx^2+s)@F"n^'(x)-(n(r+(n-1)p)x^2+(1-(-1)^n)s/2)@F"n(x)=0and 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 equationx^2(px^2+q)@F"n^''(x)+x(rx^2+s)@F"n^'(x)-(@a"nx^2+(1-(-1)^n)@b/2)@F"n(x)=0,in which @b is a free parameter and -@a"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 @b=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.