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

In a recent paper (Differential equations for generalized Jacobi polynomials, submitted for publication) Koekoek and Koekoek discovered a linear differential equation for the polynomials {Pn(x,beta.M.N) (x)}(n=0)(infinity), which are orthogonal on [-1,1] with respect to G(alpha + beta + 2)/2(alpha+beta+1) Gamma(alpha + 1)Gamma(beta + 1) (1 - x)(alpha)(1 + x)(beta) + M delta(x + 1) + N delta(x - 1), alpha,beta > - 1, M,N greater than or equal to 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. (C) 2000 Elsevier Science B.V. All rights reserved.