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 {P_n~(α,β,M,N)(x)}_(n=0)~∞, which are orthogonal on [-1, 1] with respect to (Γ(α+β+2))/(2~(α+β+1)Γ(α+1)Γ(β+1))(1-x)(1+x)~β+Mδ(x+1)+Nδ(x-1),α,β>-1, MN ≥0. (0.1) If M~2 + N~2 > 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.