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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation


Authors: Galina Filipuk and Walter Van Assche
Journal: Proc. Amer. Math. Soc. 141 (2013), 551-562
MSC (2010): Primary 34M55, 33E17; Secondary 33C47, 42C05, 65Q30
Published electronically: June 11, 2012
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Abstract | References | Similar Articles | Additional Information

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 $ _{\textup V}$ (which can be transformed to the third Painlevé equation). Initial conditions for different lattices can be transformed to the classical solutions of P $ _{\textup 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
Email: Walter.VanAssche@wis.kuleuven.be

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11468-6
PII: S 0002-9939(2012)11468-6
Keywords: Orthogonal polynomials, recurrence coefficients, Painlevé equations, Bäcklund transformations, classical solutions
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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.