Differential operators having Sobolev-type Jacobi polynomials as eigenfunctions

摘要

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 {Pnα,β,M,N(x)}n=0∞, which are orthogonal on [−1,1] with respect to (0.1)Γ(α+β+2)2α+β+1Γ(α+1)Γ(β+1)(1−x)α(1+x)β+Mδ(x+1)+Nδ(x−1),α,β>−1,M,N⩾0.If M2+N2>0 this differential equation is of finite order in the following cases: (1)M>0,N=0 and β∈{0,1,2,…}.(2)M=0,N>0 and α∈{0,1,2,…}.(3)M>0,N>0 and α,β∈{0,1,2,…}. In this paper the result will be generalized to Sobolev-type Jacobi polynomials.