Differential operators having Sobolev-type Gegenbauer polynomials as eigenfunctions

摘要

We consider the Sobolev-type Gegenbauer polynomials {Pnα,M,N(x)}n=0∞, orthogonal with respect to the inner product (f,g)=Γ(2α+2)22α+1Γ(α+1)2∫−11f(x)g(x)(1−x2)αdx+M[f(−1)g(−1)+f(1)g(1)]+N[f′(−1)g′(−1)+f′(1)g′(1)],M⩾0, N⩾0, α>−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 α is a nonnegative integer this class contains a differential operator of finite order. This is of order2ifM=N=0,2α+4ifM>0,N=0,2α+8ifM=0,N>0,4α+10ifM>0,N>0.