Finding differential equations for symmetric generalized ultraspherical polynomials by using inversion methods

Abstract

We find all spectral type differential equations satisfied by the symmetric generalized ultraspherical polynomials which are orthogonal on the interval [-1,1] with respect to the classical symmetric weight function for the Jacobi polynomials together with equal mass points at both ends of the interval of orthogonality. In order to find explicit formulas for the coefficients of these differential equations we have to solve systems of equations involving derivatives of the classical Jacobi polynomials. These systems of equations have a unique solution which is given explicitly. This is a consequence of the Jacobi inversion formula.