Geodesic
A Particular Solution of a Painlevé System in Terms of the Hypergeometric Function ${}_{n+1}F_n$
Symmetry, integrability and geometry: methods and applications, Tome 6 (2010) Cet article a éte moissonné depuis la source Math-Net.Ru

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In a recent work, we proposed the coupled Painlevé VI system with $A^{(1)}_{2n+1}$-symmetry, which is a higher order generalization of the sixth Painlevé equation ($P_{\rm{VI}}$). In this article, we present its particular solution expressed in terms of the hypergeometric function ${}_{n+1}F_n$. We also discuss a degeneration structure of the Painlevé system derived from the confluence of ${}_{n+1}F_n$.
Keywords: affine Weyl group; generalized hypergeometric functions; Painlevé equations. 
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     author = {Takao Suzuki},
     title = {A~Particular {Solution} of {a~Painlev\'e} {System} in {Terms} of the {Hypergeometric} {Function} ${}_{n+1}F_n$},
     journal = {Symmetry, integrability and geometry: methods and applications},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a77/}
}
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Takao Suzuki. A Particular Solution of a Painlevé System in Terms of the Hypergeometric Function ${}_{n+1}F_n$. Symmetry, integrability and geometry: methods and applications, Tome 6 (2010). http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a77/

[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 | DOI | MR | Zbl

[2] Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé. A modern theory of special functions, Aspects of Mathematics, E16, Friedr. Vieweg Sohn, Braunschweig, 1991 | MR | Zbl

[3] Okubo K., Takano K., Yoshida S., “A connection problem for the generalized hypergeometric equation”, Funkcial. Ekvac., 31 (1988), 483–495 | MR | Zbl

[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