The semiclassical Sobolev orthogonal polynomials: A general approach

Abstract

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, D p Dr ⟩ , 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.

On some classical type Sobolev orthogonal polynomials

“…where t 0 ≥ 1 is an arbitrary parameter. Since 1 l k,c P (α,β) k (t 0 ) > 0, we conclude that P n (α, β, c, t 0 ; x) are modified kernel polynomials of type (5). Observe that scaled by eigenvalues polynomial kernels of some Sobolev orthogonal polynomials already appeared in the literature, see [9].…”

“…The theory of Sobolev orthogonal polynomials is nowadays of a considerable interest. A brief account of the theory state can be found in [5] and a more recent survey is given in [10]. Before we shall discuss the examples of Sobolev orthogonal polynomials related to (5), we need the following definition.…”

A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case

“…This generating function can be found, for instance, in [4, p. 176] or [7, p. 175]. By using (5) it is easy to establish this result for general values of the parameter β.…”

“…Very simple manipulations of the explicit representation (5) show that the main algebraic properties of the classical Meixner polynomials still hold for the general case β ∈ R and c ∈ R \ {0, 1}, although the orthogonality given in (4) holds only for β > 0 and 0 < c < 1. In particular, for n ≥ 1, Meixner polynomials satisfy the three-term recurrence relation…”

“…This problem remains open (open problem 2 in the survey [9]). Very recently, the Δ-Meixner-Sobolev orthogonal polynomials have been used as an example in [5], where a general approach to Sobolev inner products has been made independently whether the involved measures are continuous or discrete.…”