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Zeros of varying Laguerre-Krall orthogonal polynomials

Authors: Laura Castaño–García and Juan J. Moreno–Balcázar
Journal: Proc. Amer. Math. Soc. 141 (2013), 2051-2060
MSC (2010): Primary 33C47; Secondary 42C05
Published electronically: January 17, 2013
MathSciNet review: 3034430
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Abstract: In this paper we introduce a sequence of varying orthogonal polynomials related to a Laguerre weight where this absolutely continuous measure is perturbed by a sequence of nonnegative masses located at the origin. The main objective is to obtain asymptotic relations between the zeros of these polynomials and the zeros of the Bessel functions of the first kind (or linear combinations of them). This is done through Mehler-Heine type formulas. With these relations we can easily compute asymptotically the zeros of these polynomials. We show some numerical experiments.

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  • 1. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover Publications, Inc., New York, 1972). MR 1225604 (94b:00012)
  • 2. M. Alfaro, J. J. Moreno-Balcázar, A. Peña, M. L. Rezola, A new approach to the asymptotics of Sobolev type orthogonal polynomials, J. Approx. Theory 163(4) (2011) 460-480. MR 2775140
  • 3. R. Álvarez-Nodarse, J. J. Moreno-Balcázar, Asymptotic properties of generalized Laguerre orthogonal polynomials, Indag. Math. 15(2) (2004) 151-165. MR 2071854 (2005e:33003)
  • 4. W. Gautschi, Orthogonal polynomials: Computation and Approximation (Oxford University Press, 2004). MR 2061539 (2005e:42001)
  • 5. W. Gautschi, Orthogonal polynomials (in Matlab), J. Comput. Appl. Math. 178 (2005) 215-234. MR 2127881 (2006b:33011)
  • 6. R. Koekoek, Generalizations of the classical Laguerre polynomials and some q-analogues (Thesis, Delft University of Technology, 1990). MR 2714461
  • 7. R. Koekoek, H.G. Meijer, A generalization of Laguerre polynomials, SIAM J. Math. Anal. 24(3) (1993) 768-782. MR 1215437 (94b:33007)
  • 8. A. M. Krall, Orthogonal polynomials satisfying fourth order differential equations, Proc. Roy. Soc. Edinburgh 87 (1981) 271-288. MR 606336 (82d:33021)
  • 9. H. L. Krall, On orthogonal polynomials satisfying a certain fourth order differential equation, The Pennsylvania State College Bulletin 6 (1940) 1-24. MR 0002679 (2:98a)
  • 10. P. G. Nevai, Orthogonal Polynomials, Memoirs Amer. Math. Soc., vol. 213 (Amer. Math. Soc., Providence, RI, 1979). MR 519926 (80k:42025)
  • 11. G. Szegő, Orthogonal Polynomials, vol. 23 of Amer. Math. Soc. Colloq. Publ., 4th ed. (Amer. Math. Soc., Providence, RI, 1975). MR 0310533 (46:9631)

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Additional Information

Laura Castaño–García
Affiliation: Departamento de Estadística y Matemática Aplicada, Universidad de Almería, 04120 Almería, Spain

Juan J. Moreno–Balcázar
Affiliation: Departamento de Estadística y Matemática Aplicada, Universidad de Almería, 04120 Almería, Spain

Received by editor(s): June 15, 2011
Received by editor(s) in revised form: October 2, 2011
Published electronically: January 17, 2013
Additional Notes: This research was supported by MICINN of Spain under grants MTM2008-06689-C02-01 and MTM2011-28952-C02-01, and Junta de Andalucía (FQM229 and P09–FQM–4643).
Communicated by: Walter Van Assche
Article copyright: © Copyright 2013 American Mathematical Society

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