An elementary representation of the higher-order Jacobi-type differential equation

We investigate the differential equation for the Jacobi-type polynomials which are orthogonal on the interval [−1,1] with respect to the classical Jacobi measure and an additional point mass at one endpoint. This scale of higher-order equations was introduced by J. and R. Koekoek in 1999 essentially by using special function methods. In this paper, a completely elementary representation of the Jacobi-type differential operator of any even order is given. This enables us to trace the orthogonality relation of the Jacobi-type polynomials back to their differential equation. Moreover, we establish a new factorization of the Jacobi-type operator into a product of linear second-order operators, which gives rise to a recurrence relation with respect to the order of the equation. Finally, two interrelations with the differential equation for the symmetric ultraspherical-type polynomials are pointed out.