The semiclassical Sobolev orthogonal polynomials: A general approach

We say that the polynomial sequence (Q (λ) n ) is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product 〈p, r〉S = 〈u, p r〉+ λ 〈u,DpDr〉 , where u is a semiclassical linear functional, D is the differential, the difference or the q–difference operator, and λ is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional u. The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator D considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time.