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

MSC:
34B30Special ODE (Mathieu, Hill, Bessel, etc.)
References:
[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.
[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.
[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 τ-function of the Painlevé equations, Physica,2D (1981), pp. 525–535.
[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.
[8]K.Okamoto,Introduction to the Painlevé equations, Sophia Kokyuroku in Math.,19 (1985) (in Japanese).
[9]K.Okamoto,Sur les échelles aux fonctions spéciales et l’équation de Toda, à paraítre dans J. Fac. Sci. Univ. Tokyo.
[10]P. Painlevé,Sur les équations différentielles du second ordre à points critiques fixes, Oeuvres, t. III, (1977), pp. 115–119.