Vol. 257, No. 1, 2012

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On orthogonal polynomials with respect to certain discrete Sobolev inner product

Francisco Marcellán, Ramadan Zejnullahu, Bujar Fejzullahu and Edmundo Huertas

Vol. 257 (2012), No. 1, 167–188
Abstract

In this paper we deal with sequences of polynomials orthogonal with respect to the discrete Sobolev inner product

        ∫
∞                               ′   ′
⟨f,g⟩S = 0  ω(x)f(x)g(x)dx + M f(ξ)g(ξ)+ N f(ξ)g(ξ),

where ω is a weight function, ξ 0, and M,N 0. The location of the zeros of discrete Sobolev orthogonal polynomials is given in terms of the zeros of standard polynomials orthogonal with respect to the weight function ω. In particular, for ω(x) = xαex we obtain the asymptotics for discrete Laguerre–Sobolev orthogonal polynomials.

Keywords
orthogonal polynomials, discrete Sobolev polynomials, Laguerre polynomials, asymptotics
Mathematical Subject Classification 2010
Primary: 33C47
Secondary: 42C05
Milestones
Received: 20 June 2011
Revised: 23 January 2012
Accepted: 25 January 2012
Published: 19 June 2012
Authors
Francisco Marcellán
Departamento de Matemáticas
Escuela Politécnica Superior
Universidad Carlos III de Madrid
Avenida de la Universidad 30
28911 Leganés
Spain
Ramadan Zejnullahu
Faculty of Mathematics and Sciences
University of Prishtina
Mother Teresa 5
10000 Prishtina
Kosovo
Bujar Fejzullahu
Faculty of Mathematics and Sciences
University of Prishtina
Mother Teresa 5
1000 Prishtina
Kosovo
Edmundo Huertas
Departamento de Matematicas
Escuela Politécnica Superior
Universidad Carlos III de Madrid
Avenida de la Universidad 30
28911 Leganés
Spain