The differential equation for Jacobi-Sobolev orthogonal polynomials with two linear perturbutions

We explicitly determine the higher-order differential equation for the Jacobi-Sobolev polynomials which extend the classical Jacobi polynomials with parameters alpha is an element of N0, beta > -1 by two linear perturbations. More precisely, these polynomials are orthogonal with respect to a discrete Sobolev inner product which is associated with the Jacobi measure on the interval [-1, 1] and two point masses at the right end point involving functions and their first derivatives. The corresponding Jacobi-Sobolev differential operator of order 4 alpha + 10 is appropriately defined by four elementary terms reflecting the influence of the point masses. Moreover, we prove that the new operator is symmetric with respect to the Sobolev inner product. This fundamental property readily implies the orthogonality of the Jacobi-Sobolev polynomials in the inner product space. (c) 2022 Elsevier Inc. All rights reserved.