A basic class of symmetric orthogonal functions with six free parameters

Abstract

In a previous paper, we introduced a basic class of symmetric orthogonal functions (BCSOF) by an extended theorem for Sturm–Liouville problems with symmetric solutions. We showed that the foresaid class satisfies the differential equation x 2 ( p x 2 + q ) Φ n ″ ( x ) + x ( r x 2 + s ) Φ n ′ ( x ) − ( λ n x 2 + ( 1 − ( − 1 ) n ) γ / 2 ) Φ n ( x ) = 0 , where λ n = ( n + ( θ − 1 ) ( 1 − ( − 1 ) n ) / 2 ) ( r + ( n − 1 + ( θ − 1 ) ( 1 − ( − 1 ) n ) / 2 ) p ) ; γ = θ ( s + ( θ − 1 ) q ) and contains four important sub-classes of symmetric orthogonal functions. Moreover, for θ = 1 , it is reduced to a basic class of symmetric orthogonal polynomials (BCSOP), which respectively generates the generalized ultraspherical polynomials, generalized Hermite polynomials and two other sequences of finite symmetric orthogonal polynomials. In this paper, again by using the extended theorem, we introduce a further basic class of symmetric orthogonal functions with six parameters and obtain its standard properties. We show that the new class satisfies the equation x 2 ( p x 2 + q ) Φ n ″ ( x ) + x ( r x 2 + s ) Φ n ′ ( x ) − ( a n x 2 + ( − 1 ) n c + d ) Φ n ( x ) = 0 , in which c , d are two free parameters and − a n denotes eigenvalues corresponding to the defined class. We then introduce four orthogonal sub-classes of the foresaid class and study their properties in detail. Since the introduced class is a generalization of BCSOF for − c = d = γ / 2 , the four mentioned sub-classes naturally generalize the generalized ultraspherical polynomials, generalized Hermite polynomials and two sequences of finite classical symmetric orthogonal polynomials, again.