Differential Equations for Symmetric Generalized Ultraspherical Polynomials

We look for differential equations satisfied by the generalized Jacobi polynomials { P n (x) }∞ n=0 which are orthogonal on the interval [−1, 1] with respect to the weight function Γ(α+ β + 2) 2α+β+1Γ(α+ 1)Γ(β + 1) (1− x)(1 + x) +Mδ(x+ 1) +Nδ(x− 1), where α > −1, β > −1, M ≥ 0 and N ≥ 0. In the special case that β = α and N = M we find all differential equations of the form ∞ ∑ i=0 ci(x)y (x) = 0, y(x) = P n (x), where the coefficients {ci(x)} ∞ i=1 are independent of the degree n. We show that if M > 0 only for nonnegative integer values of α there exists exactly one differential equation which is of finite order 2α+ 4. By using quadratic transformations we also obtain differential equations for the polynomials { P α,± 1 2 ,0,N n (x) }∞ n=0 for all α > −1 and N ≥ 0. AMS Subject Classification : Primary 33C45 (33A65), Secondary 34A35