Differential operators having Sobolev-type Jacobi polynomials as eigenfunctions

• Authors: H. Bavinck
• Venue: Journal of Computational and Applied Mathematics
• Year: 2003

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 Γ(α + β + 2)/2α+β+1 Γ(α + 1)Γ(β + 1)(1 - x)α(1 + x)β +Mδ(x + 1) + Nδ(x - 1),α,β, - 1, M,N ≥ 0. (0.1) 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.