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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
Published electronically: February 7, 2014
MathSciNet review: 3168479
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Abstract | References | Similar Articles | Additional Information

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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  • [1] Renato Álvarez-Nodarse and Juan J. Moreno-Balcázar, Asymptotic properties of generalized Laguerre orthogonal polynomials, Indag. Math. (N.S.) 15 (2004), no. 2, 151-165. MR 2071854 (2005e:33003),
  • [2] T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York, 1978. MR 0481884 (58 #1979)
  • [3] Herbert Dueñas and Francisco Marcellán, Laguerre-type orthogonal polynomials: electrostatic interpretation, Int. J. Pure Appl. Math. 38 (2007), no. 3, 345-358. MR 2347871 (2008i:33032)
  • [4] Herbert Dueńas, Edmundo J. Huertas, and Francisco Marcellán, Analytic properties of Laguerre-type orthogonal polynomials, Integral Transforms Spec. Funct. 22 (2011), no. 2, 107-122. MR 2749390 (2011k:33029),
  • [5] Herbert Dueñas, Edmundo J. Huertas, and Francisco Marcellán, Asymptotic properties of Laguerre-Sobolev type orthogonal polynomials, Numer. Algorithms 60 (2012), no. 1, 51-73. MR 2903969,
  • [6] Bujar Xh. Fejzullahu and Ramadan Xh. Zejnullahu, Orthogonal polynomials with respect to the Laguerre measure perturbed by the canonical transformations, Integral Transforms Spec. Funct. 21 (2010), no. 7-8, 569-580. MR 2680763 (2011i:33022),
  • [7] F. Alberto Grünbaum, Variations on a theme of Heine and Stieltjes: an electrostatic interpretation of the zeros of certain polynomials, Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997), J. Comput. Appl. Math. 99 (1998), no. 1-2, 189-194. MR 1662694 (99j:33012),
  • [8] Edmundo J. Huertas, Francisco Marcellán, and Fernando R. Rafaeli, Zeros of orthogonal polynomials generated by canonical perturbations of measures, Appl. Math. Comput. 218 (2012), no. 13, 7109-7127. MR 2880296,
  • [9] Mourad E. H. Ismail, More on electrostatic models for zeros of orthogonal polynomials, Proceedings of the International Conference on Fourier Analysis and Applications (Kuwait, 1998), Numer. Funct. Anal. Optim. 21 (2000), no. 1-2, 191-204. MR 1759996 (2001h:33002),
  • [10] Mourad E. H. Ismail, An electrostatics model for zeros of general orthogonal polynomials, Pacific J. Math. 193 (2000), no. 2, 355-369. MR 1755821 (2001i:33009),
  • [11] Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, with two chapters by Walter Van Assche, with a foreword by Richard A. Askey, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. MR 2191786 (2007f:33001)
  • [12] Roelof Koekoek, Generalizations of the classical Laguerre polynomials and some q-analogues, ProQuest LLC, Ann Arbor, MI, 1990. Thesis (Dr.)-Technische Universiteit Delft (The Netherlands). MR 2714461
  • [13] Tom H. Koornwinder, Orthogonal polynomials with weight function $ (1-x)^{\alpha }(1+x)^{\beta }+$ $ M\delta (x+1)+N\delta (x-1)$, Canad. Math. Bull. 27 (1984), 205-214. MR 740416 (85i:33011),
  • [14] G. L. Lopes, Convergence of Padé approximants for meromorphic functions of Stieltjes type and comparative asymptotics for orthogonal polynomials, Mat. Sb. (N.S.) 136(178) (1988), no. 2, 206-226, 301 (Russian); English transl., Math. USSR-Sb. 64 (1989), no. 1, 207-227. MR 954925 (90g:30003)
  • [15] G. L. Lopes, Comparative asymptotics for polynomials that are orthogonal on the real axis, Mat. Sb. (N.S.) 137(179) (1988), no. 4, 500-525, 57 (Russian); English transl., Math. USSR-Sb. 65 (1990), no. 2, 505-529. MR 981523 (90c:42030)
  • [16] F. Marcellán, A. Martínez-Finkelshtein, and P. Martínez-González, Electrostatic models for zeros of polynomials: old, new, and some open problems, J. Comput. Appl. Math. 207 (2007), no. 2, 258-272. MR 2345246 (2008h:33017),
  • [17] F. Marcellán, A. Branquinho, and J. Petronilho, Classical orthogonal polynomials: a functional approach, Acta Appl. Math. 34 (1994), no. 3, 283-303. MR 1273613 (95b:33024),
  • [18] F. Marcellán and A. Ronveaux, Differential equation for classical type orthogonal polynomials, Canad. Math. Bull. 32 (1989), 404-411. MR 1019404 (90j:33014)
  • [19] G. Szegő, Orthogonal Polynomials, $ 4^{th}$ ed., Amer. Math. Soc. Colloq. Publ. Series, vol. 23, Amer. Math. Soc., Providence, RI, 1975. MR 0372517 (51 #8724)
  • [20] Galliano Valent and Walter Van Assche, The impact of Stieltjes' work on continued fractions and orthogonal polynomials: additional material, Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), J. Comput. Appl. Math. 65 (1995), no. 1-3, 419-447. MR 1379147 (97c:33001),

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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

Francisco Marcellán Español
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganés, Madrid, Spain

Héctor Pijeira Cabrera
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganés, Madrid, Spain

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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