Differential Equations for Symmetric Generalized Ultraspherical Polynomials

Abstract

We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to the classical weight function for the Jacobi polynomials together with point masses at both endpoints. In the special symmetric case that both parameters are equal and also the two point masses are equal we find all differential equations of spectral type satisfied by these symmetric generalized ultraspherical polynomials. We show that if the point masses are positive only for nonnegative integer values of the parameter there exists exactly one differential equation of spectral type which is of finite order. By using quadratic transformations we also obtain differential equations for some related sets of generalized Jacobi polynomials. In these cases we find finite order differential equations even though one of the parameters is not equal to an integer.