The Jacobi-Sobolev, Laguerre-Sobolev, and Gegenbauer-Sobolev differential equations and their interrelations

Recently, the author determined the higher-order differential operator having the Jacobi-Sobolev polynomials as its eigenfunctions for certain eigenvalues. These polynomials form an orthogonal system with respect to an inner product equipped with the Jacobi measure on the interval [-1, 1] with parameters alpha is an element of N-0, beta > - 1 and two point masses N, S > 0 at the right end point of the interval involving functions and their first derivatives. The first purpose of the present paper is to reveal how the Jacobi-Sobolev equation reduces to the differential equation satisfied by the Laguerre-Sobolev polynomials on the positive half line via a confluent limiting process as beta -> infinity. Secondly, we explicitly establish the differential equation for the symmetric Gegenbauer-Sobolev polynomials by employing a quadratic transformation of the argument. Each of the three differential operators involved is of order 4 alpha + 10 and symmetric with respect to the corresponding Sobolev inner product.