New representation and factorizations of the higher-order ultraspherical-type differential equations

The paper deals with the class of linear differential equations of any even order 2α+42α+4, α∈N0α∈N0, which are associated with the so-called ultraspherical-type polynomials. These polynomials form an orthogonal system on the interval [−1,1][−1,1] with respect to the ultraspherical weight function (1−x2)α(1−x2)α and additional point masses of equal size at the two endpoints. The differential equations of “ultraspherical-type” were developed by R. Koekoek in 1994 by utilizing special function methods. In the present paper, a new and completely elementary representation of these higher-order differential equations is presented. This result is used to deduce the orthogonality relation of the ultraspherical-type polynomials directly from the differential equation property. Moreover, we introduce two types of factorizations of the corresponding differential operators of order 2α+42α+4 into a product of α+2α+2 linear second-order operators.