An electrostatic model for zeros of perturbed Laguerre polynomials
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- by Edmundo J. Huertas Cejudo, Francisco Marcellán Español and Héctor Pijeira Cabrera PDF
- Proc. Amer. Math. Soc. 142 (2014), 1733-1747 Request permission
Abstract:
In this paper we consider the sequences of polynomials $\{Q_{n}^{(\alpha )}\}_{n\geq 0}$, orthogonal with respect to the inner product \[ \langle f,g\rangle _{\nu }=\int _{0}^{+\infty }f(x)g(x)d\mu (x)+\sum _{j=1}^{m}a_{j} f(c_{j})g(c_{j}), \] 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.References
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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
- MR Author ID: 921764
- ORCID: 0000-0001-6802-3303
- 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
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3168479