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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation
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by Galina Filipuk and Walter Van Assche PDF
Proc. Amer. Math. Soc. 141 (2013), 551-562 Request permission

Abstract:

We investigate generalizations of the Charlier polynomials on the lattice $\mathbb {N}$, on the shifted lattice $\mathbb {N}+1-\beta$, and on the bi-lattice $\mathbb {N}\;\cup$ $(\mathbb {N}+1-\beta )$. We show that the coefficients of the three-term recurrence relation for the orthogonal polynomials are related to solutions of the fifth Painlevé equation P$_{\mathrm {V}}$ (which can be transformed to the third Painlevé equation). Initial conditions for different lattices can be transformed to the classical solutions of P$_{\mathrm {V}}$ with special values of the parameters.
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Additional Information
  • Galina Filipuk
  • Affiliation: Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, Warsaw 02-097, Poland
  • Email: filipuk@mimuw.edu.pl
  • Walter Van Assche
  • Affiliation: Department of Mathematics, KU Leuven, Celestijnenlaan 200B, Box 2400, BE-3001 Leuven, Belgium
  • MR Author ID: 176825
  • ORCID: 0000-0003-3446-6936
  • Email: Walter.VanAssche@wis.kuleuven.be
  • Received by editor(s): June 15, 2011
  • Received by editor(s) in revised form: June 30, 2011
  • Published electronically: June 11, 2012
  • Additional Notes: The first author is partially supported by Polish MNiSzW Grant N N201 397937.
    The second author was supported by the Belgian Interuniversity Attraction Pole P6/02, FWO Grant G.0427.09 and KU Leuven Research Grant OT/08/033.
  • Communicated by: Sergei K. Suslov
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 551-562
  • MSC (2010): Primary 34M55, 33E17; Secondary 33C47, 42C05, 65Q30
  • DOI: https://doi.org/10.1090/S0002-9939-2012-11468-6
  • MathSciNet review: 2996960