Differential operators having Sobolev-type Jacobi polynomials as eigenfunctions

In a recent paper Koekoek and Koekoek (J. Comput. Appl. Math. 126 (2000) 1-31) discovered a linear differential equation for the Jacobi-type polynomials {P-n(alpha,beta,M,N)(x)}(n=0)(infinity),which are orthogonal on [-1,1] with respect to [Gamma(alpha+beta+2) / 2(alpha+beta+1)Gamma(alpha + 1)Gamma(beta+1)] [(1-x)(alpha)(1+x)(beta)] +Mdelta(x-1)+Ndelta(x-1),alpha,beta > -1, M, N greater than or equal to 0. If M-2 + N-2 > 0 this differential equation is of finite order in the following cases: (1) M > 0, N = 0 and beta is an element of {0, 1, 2....}. (2) M = 0, N > 0 and alpha is an element of {0, 1, 2,...}. (3) M > 0, N > 0 and alpha, beta is an element of {0, 1, 2,...}. In this paper the result will be generalized to Sobolev-type Jacobi polynomials. (C) 2003 Elsevier Science B.V. All rights reserved.