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Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 6 (2010), 078, 11 pages      arXiv:1004.0059      http://dx.doi.org/10.3842/SIGMA.2010.078
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

A Particular Solution of a Painlevé System in Terms of the Hypergeometric Function n+1Fn

Takao Suzuki
Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

Received June 23, 2010, in final form September 29, 2010; Published online October 07, 2010

Abstract
In a recent work, we proposed the coupled Painlevé VI system with A2n+1(1)-symmetry, which is a higher order generalization of the sixth Painlevé equation (PVI). In this article, we present its particular solution expressed in terms of the hypergeometric function n+1Fn. We also discuss a degeneration structure of the Painlevé system derived from the confluence of n+1Fn.

Key words: affine Weyl group; generalized hypergeometric functions; Painlevé equations.

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References

  1. Fuji K., Suzuki T., Drinfeld-Sokolov hierarchies of type A and fourth order Painlevé systems, Funkcial. Ekvac. 53 (2010), 143-167, arXiv:0904.3434.
  2. Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé. A modern theory of special functions, Aspects of Mathematics, Vol. E16, Friedr. Vieweg & Sohn, Braunschweig, 1991.
  3. Okubo K., Takano K., Yoshida S., A connection problem for the generalized hypergeometric equation, Funkcial. Ekvac. 31 (1988), 483-495.
  4. Suzuki T., A class of higher order Painlevé systems arising from integrable hierarchies of type A, arXiv:1002.2685.
  5. Sakai H., Private communication.
  6. Tsuda T., UC hierarchy and monodromy preserving deformation, arXiv:1007.3450.

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