Hostname: page-component-7dd5485656-zklqj Total loading time: 0 Render date: 2025-10-30T20:01:35.009Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the transcendency of Painlevé’s first transcendent

Published online by Cambridge University Press:  22 January 2016

Keiji Nishioka
Keiji Nishioka*
Affiliation:
Takabatake-cho 184-632, Nara, 630, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Here we shall prove that Painlevé’s first transcendent, a solution of the equation y″ = 6y2 + x, can not be described as any combination of solutions of first order algebraic differential equations and those of linear differential equations. This result gives an answer to the question whether the function is truely new or not.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 109 , March 1988 , pp. 63 - 67
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

[1] Bialynicki-Birula, A., On Galois theory of fields with operators, Amer. J. Math., 84 (1962), 89–109.CrossRefGoogle Scholar
[1] Kolchin, E. R., Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.Google Scholar
[1] Liouville, R., Sur les équations différentielles du second order à points critiques fixes, Oevres de P. Painlevé, vol. 3 (1975), pp. 81–83.Google Scholar
[1] Nishioka, K., A class of transcendental functions containing elementary and elliptic ones, Osaka J. Math., 22 (1985), 743–753.Google Scholar
[1] Okamoto, K., Introduction to the Painlevé Equations (in Japanese), Sophia Kokyuroku in Math., vol. 19, 1985.Google Scholar
[1] Pommaret, J. F., Differential Galois theory, Gordon and Breach, 1983.Google Scholar
[1] Umemura, H., Birational automorphism groups and differential equations, to appear.Google Scholar