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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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


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:
  • 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:
  • 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:
  • MathSciNet review: 2996960