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Studies of the Painlevé equations. I: Sixth Painlevé equation $P\sb{VI}$. (English) Zbl 0637.34019
The author studies birational canonical transformations of the Hamiltonian system ${\frak H}$ associated with the sixth Painlevé equation. Solutions of ${\frak H}$, which can be written by use of hypergeometric functions are studied in details.
Reviewer: P.Bolley

34B30Special ODE (Mathieu, Hill, Bessel, etc.)
Full Text: DOI
[1] N.Bourbaki,Groupes et Algèbres de Lie, Chapitres 4, 5 et 6, Masson, Paris.
[2] M. Jimbo -T. Miwa,Monodromy preserving deformation of linear ordinary differential equations with rational coefficients, II, Physica,2D (1981), pp. 407--448. · Zbl 1194.34166
[3] N. A. Lukashevich,The Theory of Painlevé’s equations, Différents, Uravneniya,6 (1970), pp. 329--333.
[4] K. Okamoto,Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Jap. J. Math.,5 (1979), pp. 1--79. · Zbl 0426.58017
[5] K. Okamoto,Polynomial Hamiltonians associated with Painlevé equations, I, Proc. Japan Acad.,56, Ser. A (1980), pp. 264--268; II, ibid., pp. 367--371. · Zbl 0476.34010 · doi:10.3792/pjaa.56.264
[6] K. Okamoto,On the {$\tau$}-function of the Painlevé equations, Physica,2D (1981), pp. 525--535. · Zbl 1194.34171
[7] K. Okamoto,Isomonodromic deformation and Painlevé equations, and the Garnier system. J. Fac. Sci. Univ. Tokyo Sect. IA Math.,33 (1986), pp. 575--618. · Zbl 0631.34011
[8] K.Okamoto,Introduction to the Painlevé equations, Sophia Kokyuroku in Math.,19 (1985) (in Japanese). · Zbl 0563.34035
[9] K.Okamoto,Sur les échelles aux fonctions spéciales et l’équation de Toda, à paraítre dans J. Fac. Sci. Univ. Tokyo. · Zbl 0696.33004
[10] P. Painlevé,Sur les équations différentielles du second ordre à points critiques fixes, Oeuvres, t. III, (1977), pp. 115--119.