Differential operators having Sobolev-type Gegenbauer polynomials as eigenfunctions

We consider the Sobolev-type Gegenbauer polynomials {p(n)(x,M,N)(x)}(n=0)(infinity), orthogonal with respect to the inner product ( f,g) Gamma(2 alpha + 2)/2(2 alpha+1)Gamma(alpha+1)(2) integral(-1)(1)f(x)g(x)(1-x(2))(alpha)dx +M[f(-1)g(-1)+f(1)g(1)]+N[f'(-1)g'(-1)+f'(1)g'(1)], M greater than or equal to 0, N greater than or equal to 0, alpha > -1. It is the purpose of this paper to show that these polynomials are eigenfunctions of a class of linear differential operators, usually of infinite order. In the case that a is a nonnegative integer this class contains a differential operator of finite order. This is of order 2 if M = N = 0, 2 alpha + 4 if M> 0, N = 0, 2 alpha + 8 if M = 0, N > 0, 4 alpha + 10 if M > 0. N > 0 (C) 2000 Elsevier Science B.V. All rights reserved. MSC: 33C45; 34A35.