# Strong and Plancherel-Rotach asymptotics of non-diagonal Laguerre-Sobolev orthogonal polynomials

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 dc.contributor.author Marcellán Español, Francisco José dc.contributor.author Moreno Balcázar, Juan José dc.date.accessioned 2009-12-16T12:01:25Z dc.date.available 2009-12-16T12:01:25Z dc.date.issued 2001-05 dc.identifier.bibliographicCitation Journal of Approximation Theory, 2001, vol. 110, n. 1, p. 54-73 dc.identifier.issn 0021-9045 dc.identifier.uri http://hdl.handle.net/10016/6118 dc.description 20 pages, no figures.-- MSC2000 code: 42C05. dc.description MR#: MR1826085 (2002f:42024) dc.description Zbl#: Zbl 0983.42013 dc.description.abstract We study properties of the monic polynomials $\{Q_n\}_{n\in\bbfN}$ orthogonal with respect to the Sobolev inner product $$(p,q)_S= \int infty_0 (p,p')\pmatrix 1 & \mu\\ \mu &\lambda\endpmatrix \pmatrix q\\ q'\endpmatrix x alpha e -x} dx,$$ where $\lambda- \mu > 0$ and $\alpha> -1$. This inner product can be expressed as $$(p,q)_S= \int infty_0 p(x) q(x)((\mu+ 1) x- \alpha\mu) x \alpha- 1} e -x} dx+ \lambda\int infty_0 p'q' x alpha e -x} dx,$$ when $\alpha> 0$. In this way, the measure which appears in the first integral is not positive on $[0,\infty)$ for $\mu\in \bbfR\setminus[- 1,0]$. The aim of this paper is the study of analytic properties of the polynomials $Q_n$. First, we give an explicit representation for $Q_n$ using an algebraic relation between Sobolev and Laguerre polynomials together with a recursive relation for $\widetilde k_n= (Q_n,Q_n)_S$. Then we consider analytic aspects. We first establish the strong asymptotics of $Q_n$ on $\bbfC\setminus[0,\infty)$ when $\mu\in \bbfR$ and we also obtain an asymptotic expression on the oscillatory region, that is, on $(0,\infty)$. Then we study the Plancherel-Rotach asymptotics for the Sobolev polynomials $Q_n(nx)$ on $\bbfC\setminus[0, 4]$ when $\mu\in (- 1,0]$. As a consequence of these results we obtain the accumulation sets of zeros and of the scaled zeros of $Q_n$. We also give a Mehler-Heine type formula for the Sobolev polynomials which is valid on compact subsets of $\bbfC$ when $\mu\in (-1,0]$, and hence in this situation we obtain a more precise result about the asymptotic behaviour of the small zeros of $Q_n$. This result is illustrated with three numerical examples. dc.description.sponsorship Research of first author (F.M.) supported by Dirección General de Enseñanza Superior (DGES) of Spain under Grant PB96-0120-C03-01 and INTAS 93-219 Ext. Research of second author (J.J.M.B.) partially supported by Junta de Andalucía, Grupo de Investigación FQM 0229. dc.format.mimetype application/pdf dc.language.iso eng dc.publisher Elsevier dc.rights © Elsevier dc.subject.other Sobolev orthogonal polynomials dc.subject.other Laguerre polynomials dc.subject.other Bessel functions dc.subject.other Scaled polynomials dc.subject.other Asymptotics dc.subject.other Plancherel-Rotach asymptotics dc.title Strong and Plancherel-Rotach asymptotics of non-diagonal Laguerre-Sobolev orthogonal polynomials dc.type article dc.type.review PeerReviewed dc.description.status Publicado dc.relation.publisherversion http://dx.doi.org/10.1006/jath.2000.3530 dc.subject.eciencia Matemáticas dc.identifier.doi 10.1006/jath.2000.3530 dc.rights.accessRights openAccess
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