We look for spectral type differential equations satisfied by the generalized Jacobi polynomials, which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with two point masses at the endpoints of the interval of orthogonality. We show that such a differential equation is uniquely determined and we give explicit representations for the coefficients. In case of nonzero mass points the order of this differential equation is infinite, except for nonnegative integer values of (one of) the parameters. Otherwise, the finite order is explictly given in terms of the parameters.Comment: 33 pages, submitted for publicatio

Abstract.Koornwinder's generalized Laguerre polynomials {L°' (x)}^=0 are orthogonal on the interval [0, oo) with respect to the weight function r,'+|.xae~j: + Nô(x), a > -1 , N > 0. We show that these polynomials for N > 0 satisfy a unique differential equation of the form

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Abstract. The Sobolev-type Laguerre polynomials {L α,M,N n (x)} ∞ n=0 are orthogonal with respect to the inner product,In 1990 the first and second author showed that in the case M > 0 and N = 0 the polynomials are eigenfunctions of a unique differential operator of the formwhereare independent of n. This differential operator is of order 2α + 4 if α is a nonnegative integer, and of infinite order otherwise.In this paper we construct all differential equations of the formwhere the coefficientsare independent of n and the coefficients a 0 (x), b 0 (x) and c 0 (x) are independent of x, satisfied by the Sobolev-type Laguerre polynomials {L α,M,N n (x)} ∞ n=0 . Further, we show that in the case M = 0 and N > 0 the polynomials are eigenfunctions of a linear differential operator, which is of order 2α + 8 if α is a nonnegative integer and of infinite order otherwise.Finally, we show that in the case M > 0 and N > 0 the polynomials are eigenfunctions of a linear differential operator, which is of order 4α + 10 if α is a nonnegative integer and of infinite order otherwise.

We look for differential equations satisfied by the generalized Jacobi polynomials P α,β,M,N n (x) ∞ n=0 which are orthogonal on the interval [−1, 1] with respect to the weight functionwhere α > −1, β > −1, M ≥ 0 and N ≥ 0.In order to find explicit formulas for the coefficients of these differential equations we have to solve systems of equations of the form