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A note on the transcendency of Painlevé’s first transcendent. (English) Zbl 0613.34030

In this note it is proved that Painlevé’s first transcendent, a solution of the equation \(y''=6y^ 2+x\), cannot be described as any combination of solutions of first order algebraic differential equations and of linear differential equations. This result puts an end to the historical controversy held between P. Painlevé and R. Liouville, whether the function is truly “new” or not.

MSC:

34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
12H05 Differential algebra
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References:

[1] Oevres de P. Painlevé 3 pp 81– (1975)
[2] Differential Algebra and Algebraic Groups (1973)
[3] Osaka J. Math 22 pp 743– (1985)
[4] Sophia Kokyuroku in Math 19 (1985)
[5] DOI: 10.2307/2372805 · Zbl 0113.03203
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