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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
MathSciNet review: 2996960
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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.

References [Enhancements On Off] (What's this?)

  • 1. Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
  • 2. V. È. Adler, Nonlinear chains and Painlevé equations, Phys. D 73 (1994), no. 4, 335–351. MR 1280883, 10.1016/0167-2789(94)90104-X
  • 3. Lies Boelen, Galina Filipuk, and Walter Van Assche, Recurrence coefficients of generalized Meixner polynomials and Painlevé equations, J. Phys. A 44 (2011), no. 3, 035202, 19. MR 2749070, 10.1088/1751-8113/44/3/035202
  • 4. T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. MR 0481884
  • 5. G. Filipuk, W. Van Assche, and L. Zhang The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation, arXiv:1105.5229v1 [math.CA]
  • 6. G. Filipuk and W. Van Assche, Recurrence coefficients of a new generalization of the Meixner polynomials, SIGMA Symmetry Integrability Geom. Methods Appl.7 (2011), 068, 11 pages.
  • 7. A. S. Fokas and M. J. Ablowitz, On a unified approach to transformations and elementary solutions of Painlevé equations, J. Math. Phys. 23 (1982), no. 11, 2033–2042. MR 679998, 10.1063/1.525260
  • 8. Valerii I. Gromak, Ilpo Laine, and Shun Shimomura, Painlevé differential equations in the complex plane, de Gruyter Studies in Mathematics, vol. 28, Walter de Gruyter & Co., Berlin, 2002. MR 1960811
  • 9. M. N. Hounkonnou, C. Hounga, and A. Ronveaux, Discrete semi-classical orthogonal polynomials: generalized Charlier, J. Comput. Appl. Math. 114 (2000), no. 2, 361–366. MR 1737084, 10.1016/S0377-0427(99)00275-7
  • 10. Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786
  • 11. Jürgen Moser, Finitely many mass points on the line under the influence of an exponential potential–an integrable system, Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974) Springer, Berlin, 1975, pp. 467–497. Lecture Notes in Phys., Vol. 38. MR 0455038
  • 12. Masatoshi Noumi, Painlevé equations through symmetry, Translations of Mathematical Monographs, vol. 223, American Mathematical Society, Providence, RI, 2004. Translated from the 2000 Japanese original by the author. MR 2044201
  • 13. Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
  • 14. C. Smet and W. Van Assche, Orthogonal polynomials on a bi-lattice, Constr. Approx., DOI:10.1007/S00365-011-9145-8, arXiv:1101.1817v1 [math.CA]
  • 15. T. Tokihiro, B. Grammaticos, and A. Ramani, From the continuous 𝑃_{𝑉} to discrete Painlevé equations, J. Phys. A 35 (2002), no. 28, 5943–5950. MR 1930545, 10.1088/0305-4470/35/28/312
  • 16. V.V. Tsegelnik, The Painlevé type equations: analytical properties of solutions and their applications, Habilitation thesis, Minsk, 2001 (in Russian).
  • 17. Walter Van Assche and Mama Foupouagnigni, Analysis of non-linear recurrence relations for the recurrence coefficients of generalized Charlier polynomials, J. Nonlinear Math. Phys. 10 (2003), no. suppl. 2, 231–237. MR 2063533, 10.2991/jnmp.2003.10.s2.19

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Additional Information

Galina Filipuk
Affiliation: Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, Warsaw 02-097, Poland

Walter Van Assche
Affiliation: Department of Mathematics, KU Leuven, Celestijnenlaan 200B, Box 2400, BE-3001 Leuven, Belgium

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.