Symmetric differential operators for Sobolev orthogonal polynomials of Laguerre- and Jacobi-type

Abstract

ABSTRACT The Laguerre-Sobolev polynomials form an orthogonal polynomial system on the positive half-line with respect to the classical Laguerre measure with parameter and, in general, two point masses , S>0 at the origin involving functions and their first derivatives. Moreover we consider the Jacobi-Sobolev polynomials on the interval with Jacobi parameters and one Sobolev point mass S>0 at the right end-point of the domain. For , both polynomial systems are known to arise as eigenfunctions of certain spectral differential operators of finite order , provided that N = 0 in the Laguerre case. In this paper, new representations of the two differential operators are established which consist of a number of elementary components reflecting the influence of the parameters, appropriately. In particular, we show that the operators are symmetric with respect to the respective Sobolev inner products and so recover the orthogonality of the eigenfunctions.