Nishioka, Keiji A note on the transcendency of Painlevé’s first transcendent. (English) Zbl 0613.34030 Nagoya Math. J. 109, 63-67 (1988). 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. Cited in 4 ReviewsCited in 21 Documents 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 Keywords:Painlevé’s first transcendent; first order algebraic differential equations; second order differential equation × Cite Format Result Cite Review PDF Full Text: DOI 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 · doi:10.2307/2372805 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.