A note on the Koekoeks’ differential equation for generalized Jacobi polynomials

摘要

In a recent paper (Differential equations for generalized Jacobi polynomials, submitted for publication) Koekoek and Koekoek discovered a linear differential equation for the 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.This differential equation is of infinite order, except in a number of cases. It is the purpose of this note to reprove and interpret the results of the Koekoeks in the finite-order cases in a short and easy way.