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An electrostatic model for zeros of perturbed Laguerre polynomials


Authors: Edmundo J. Huertas Cejudo, Francisco Marcellán Español and Héctor Pijeira Cabrera
Journal: Proc. Amer. Math. Soc. 142 (2014), 1733-1747
MSC (2010): Primary 33C45, 33C47; Secondary 42C05, 34A05
DOI: https://doi.org/10.1090/S0002-9939-2014-11968-X
Published electronically: February 7, 2014
MathSciNet review: 3168479
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Abstract: In this paper we consider the sequences of polynomials $ \{Q_{n}^{(\alpha )}\}_{n\geq 0}$, orthogonal with respect to the inner product

$\displaystyle \langle f,g\rangle _{\nu }=\int _{0}^{+\infty }f(x)g(x)d\mu (x)+\sum _{j=1}^{m}a_{j}\,f(c_{j})g(c_{j}), $

where $ d\mu (x)=x^{\alpha }e^{-x}$ is the Laguerre measure on $ \mathbb{R}_{+}$, $ \alpha \! >\!-1$, $ c_{j}\!<0$, $ a_{j}\!>0$ and $ f,\,g$ are polynomials with real coefficients. We first focus our attention on the representation of these polynomials in terms of the standard Laguerre polynomials. Next we find the explicit formula for their outer relative asymptotics, as well as the holonomic equation that such polynomials satisfy. Finally, an electrostatic interpretation of their zeros in terms of a logarithmic potential is presented.

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Additional Information

Edmundo J. Huertas Cejudo
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganés, Madrid, Spain
Address at time of publication: CMUC and Department of Mathematics, University of Coimbra, Apartado 2008, EC Santa Cruz, 3001-501, Coimbra, Portugal
Email: ehuertasce@gmail.com, ehuertas@math.uc3m.es

Francisco Marcellán Español
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganés, Madrid, Spain
Email: pacomarc@ing.uc3m.es

Héctor Pijeira Cabrera
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganés, Madrid, Spain
Email: hpijeira@math.uc3m.es

DOI: https://doi.org/10.1090/S0002-9939-2014-11968-X
Keywords: Orthogonal polynomials, electrostatic interpretation
Received by editor(s): December 22, 2011
Received by editor(s) in revised form: May 24, 2012, and June 22, 2012
Published electronically: February 7, 2014
Additional Notes: The authors were supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain, under grant MTM2009-12740-C03-01
Communicated by: Walter Van Assche
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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