On the differential equation for the Laguerre–Sobolev polynomials

Abstract

The Laguerre–Sobolev polynomials form an orthogonal polynomial system with respect to a Sobolev-type inner product associated with the Laguerre measure on the positive half-axis and two point masses M,N>0 at the origin involving functions and derivatives. These polynomials have attracted much interest over the last two decades, since they became known to satisfy, for any value of the Laguerre parameter α∈N0, a spectral differential equation of finite order 4α+10. In this paper we establish a new explicit representation of the corresponding differential operator which consists of a number of elementary components depending on α,M,N. Their interaction reveals a rich structure both being useful for applications and as a model for further investigations in the field. In particular, the Laguerre–Sobolev differential operator is shown to be symmetric with respect to the Sobolev-type inner product.