Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T10:00:49.238Z Has data issue: false hasContentIssue false
  • English
  • Français

On the rational solutions of q-Painlevé V equation

Published online by Cambridge University Press:  22 January 2016

Tetsu Masuda
Tetsu Masuda*
Affiliation:
Department of Mathematics, Kobe University, Rokko, Kobe, 657-8501, Japan, masuda@math.kobe-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give an explicit determinant formula for a class of rational solutions of a q-analogue of the Painlevé V equation. The entries of the determinant are given by the continuous q-Laguerre polynomials.

Keywords

33E1734M55
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 169 , 2003 , pp. 119 - 143
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2003

References

[1] Grammaticos, B. and Ramani, A., Discrete Painlevé equations: coalescences, limits and degeneracies, Phyica, A228 (1996), 160–171.CrossRefGoogle Scholar
[2] Grammaticos, B., Ramani, A. and Papageorgiou, V. G., Do integrable mappings have the Painlevé property, Phys. Rev. Lett., 67 (1991), 1825–1828.CrossRefGoogle Scholar
[3] Kajiwara, K., Noumi, M. and Yamada, Y., A Study on the fourth q-Painlevé Equation, J. Phys. A: Math. Gen., 34 (2001), 8563–8581.CrossRefGoogle Scholar
[4] Kajiwara, K., Noumi, M. and Yamada, Y., Discrete dynamical systems with symmetry, Lett. Math. Phys., 60 (2002), 211–219.CrossRefGoogle Scholar
[5] Kajiwara, K., Noumi, M. and Yamada, Y., private communication.Google Scholar
[6] Koekoek, R. and Swarttouw, R. F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Delft University of Technology, Department of Technical Mathematics and Informatics Report no. 98–17 (1998).Google Scholar
[7] Koike, K., On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters, Adv. Math., 74 (1989), 57–86.CrossRefGoogle Scholar
[8] Masuda, T., Ohta, Y. and Kajiwara, K., A determinant formula for a class of rational solutions of Painlevé V equation, Nagoya Math. J., 168 (2002), 1–25.CrossRefGoogle Scholar
[9] Noumi, M., Okada, S., Okamoto, K., and Umemura, H., Special polynomials associated with the Painleve equations II, Proceedings of the Taniguchi Symposium, 1997, Integrable Systems and Algebraic Geometry (Saito, M. H., Shimizu, Y., Ueno, K., eds.), Singapore: World Scientific (1998), pp. 349–372.Google Scholar
[10] Noumi, M. and Yamada, Y., Higher order Painlevé equations of type , Funkcial. Ekvac., 41 (1998), 483–503.Google Scholar
[11] Okamoto, K., Studies on the Painlevé equations III, second and fourth Painlevé equations, PII and PIV , Math. Ann., 275 (1986), 222–254.CrossRefGoogle Scholar
[12] Ramani, A., Grammaticos, B. and Hietarinta, J., Discrete versions of the Painlevé equations, Phys. Rev. Lett., 67 (1991), 1829–1832.CrossRefGoogle Scholar
[13] Sakai, H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Commun. Math. Phys., 220 (2001), no. 1, 165–229.Google Scholar
[14] Tokihiro, T., Takahashi, D., Matsukidaira, J. and Satsuma, J., From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett., 76 (1996), 3247–3250.CrossRefGoogle Scholar
[15] Taneda, M., Polynomials associated with an algebraic solution of the sixth Painlevé equation, to appear in Jap. J. Math., 27 (2001).CrossRefGoogle Scholar
[16] Umemura, H., Special polynomials associated with the Painlevé equations I, preprint.Google Scholar