On the Krall-type polynomials
type: Item
instance of: scholarly article
title: On the Krall-type polynomials
zbMATH Open document ID: 1080.33008
DOI: 10.1155/S1110757X04312076
published in: Journal of Applied Mathematics
publication date: 2005-10-31T00:00:00Z
full work available at URL: https://eudml.org/doc/52041
review text: Let \(u\) be a quasi-definite linear functional in the vector space \(\mathbb{P}\) of polynomials with complex coefficients. Perturbations of such functionals \(u\) via the addition of Dirac delta functionals \(\delta\) and their derivatives \(\delta^{\prime}\) have been the subject of extensive study in recent years. In this paper, the authors study perturbed linear functionals of the form \(\tilde{u}=u+ A\delta(x-a)-B\delta^{\prime}(x-b),\;\;a\neq b\). In particular, they give necessary and sufficient conditions for the quasi-definiteness of such a \(\tilde{u}\). In this case, the coefficients of the three-term recurrence relation that the corresponding sequence of monic orthogonal polynomials satisfies is obtained. In the case of a semiclassical linear functional \(u\), it is shown that the corresponding functional \(\tilde{u}\) is also semiclassical. This can be read in terms of the holonomic second-order linear differential equation that such polynomials satisfy. The explicit expression of their coefficients is given. Information on the location of zeros of these polynomials is also derived. The paper also provides background material on the subject as well as an extensive list of relevant papers.
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 47A06
zbMATH DE Number: 2220772
zbMATH Keywords: quasi-definite linear functionals
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Krall-type polynomials
zbMATH Keywords: addition of delta Dirac masses
zbMATH Keywords: perturbed linear functionals
author: Renato Álvarez-Nodarse
author: Jorge Arvesú
author: Francisco Marcellán
MaRDI profile type: Publication
description: scientific article