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Classical solutions of the third Painlevé equation

Published online by Cambridge University Press:  22 January 2016

Yoshihiro Murata
Yoshihiro Murata*
Affiliation:
Faculty of Economics, Nagasaki University, 4-2-1, Katafuchi, Nagasaki-shi 850, Japan
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The big problem “Do Painlevé equations define new functions?”, what is called the problem of irreducibilities of Painlevé equations, was essentially solved by H. Umemura [16], [17] and K. Nishioka [9].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 139 , September 1995 , pp. 37 - 65
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1995

References

[ 1 ] Gromak, V. I., Solutions of the third Painlevé equation, Differential Equations, 9 (1973), 1599–1600.Google Scholar
[ 2 ] Gromak, V. I., Theory of Painlevé’s equation, Differential Equations, 11 (1975), 285–287.Google Scholar
[ 3 ] Gromak, V. I., One-parameter systems of solutions of Painlevé’s equations, Differential Equations, 14 (1978), 1510–1513.Google Scholar
[ 4 ] Gromak, V. I. and Lukashevich, N. A., Special classes of solutions of Painlevé’s equations, Differential Equations, 18 (1982), 317–326.Google Scholar
[ 5 ] Gromak, V. I., Reducibility of Painlevé equations, Differential Equations, 20 (1984), 1191–1198.Google Scholar
[ 6 ] Lukashevich, N. A., On the theory of the third Painlevé equation, Differential Equations, 3 (1967), 994–999.Google Scholar
[ 7 ] Murata, Y., Rational solutions of the second and the fourth Painlevé equations, Funkcial. Ekvac, 28 (1985), 1–32.Google Scholar
[ 8 ] Murata, Y., On the irreducibility of the third Painlevé equation, in preparation.Google Scholar
[ 9 ] Nishioka, K., A note on the transcendency of Painlevé’s first transcendent, Nagoya Math. J., 109 (1988), 63–67.CrossRefGoogle Scholar
[10] Noumi, M., private communication.Google Scholar
[11] Okamoto, K., Polynomial Hamiltonians associated with Painlevé equations. I, Proc. Japan Acad. Ser. A Math. Sci., 56 (1980), 264–268; II, ibid., 367–371.Google Scholar
[12] Okamoto, K., Isomonodromic deformation and Painlevé equations, and the Gamier system, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 33 (1986), 575–618.Google Scholar
[13] Okamoto, K., Studies on the Painlevé equations IV. Third Painlevé equation PIII , Funkcial. Ekvac, 30 (1987), 305–332.Google Scholar
[14] Okamoto, K., private communication.Google Scholar
[15] Painlavé, P., Leçons de Stockholm, in “Œuvres de P. Painlevé I”, Editions du C. N. R. S., Paris, 1972, 199–818.Google Scholar
[16] Umemura, H., Birational automorphism groups and differential equations, Nagoya Math. J., 119 (1990), 1–80.CrossRefGoogle Scholar
[17] Umemura, H., On the irreducibility of the first differential equation of Painlevé, in “Algebraic geometry and commutative algebra in honor of Masayoshi Nagata”, Kinokuniya, Tokyo, 1987, 771–789.Google Scholar