A basic class of symmetric orthogonal functions with six free parameters
A basic class of symmetric orthogonal functions with six parameters is introduced. It is obtained by using an extended theorem for Sturm-Liouville problems, and its standard properties are deduced. Indeed, it is shown that the new class satisfies the differential equation \[ x^2(p x^2+q) \varphi_n''(x)+ x(r x^2+s)\varphi_n'(x)-(a_nx^2+ (-1)^nc +d)\varphi_n(x)=0, \] in which \(c,d\) are two free parameters and \(-a_n\) denotes eigenvalues corresponding to the defined class. Four orthogonal sub-classes are introduced and a detailed study of their properties is given. These four sub-classes generalize the generalized ultraspherical polynomials, the generalized Hermite polynomials and two sequences of finite classical symmetric orthogonal polynomials with weight functions \(x^{-2a}(1+x^2)^{-b}\) and \(x^{-2a}\exp(\frac{-1}{x^2})\) on \((- \infty, \infty)\), respectively.
extended Sturm-Liouville theorem for symmetric functions
two sequences of finite classical symmetric orthogonal polynomials
scientific article; zbMATH DE number 5696582
scientific article