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

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 formx2(px2+q)Φn″(x)+x(rx2+s)Φn′(x)-(n(r+(n-1)p)x2+(1-(-1)n)s/2)Φn(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 equationx2(px2+q)Φn″(x)+x(rx2+s)Φn′(x)-(αnx2+(1-(-1)n)β/2)Φn(x)=0,in which ββ is a free parameter and -αn-αn 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=sβ=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.