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2013 Frequency of $a$-points for the fifth and the third Painlevé transcendents in a sector
Shun Shimomura
Tohoku Math. J. (2) 65(4): 591-605 (2013). DOI: 10.2748/tmj/1386354297


For the fifth Painlevé transcendents in a sector, under the condition that the values taken along some curve tending to infinity are bounded away from 1 and another specified complex number, we present new upper estimates for the number of $a$-points including poles and for the growth order. As far as we are concerned with the known asymptotic solutions of the fifth Painlevé equation, this condition is easily checked, and our results are applicable to almost all of them. About concrete examples we discuss the frequency of $a$-points, the equi-distribution property and the growth order. Our method works on the third Painlevé transcendents as well, yielding an analogous result.


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Shun Shimomura. "Frequency of $a$-points for the fifth and the third Painlevé transcendents in a sector." Tohoku Math. J. (2) 65 (4) 591 - 605, 2013.


Published: 2013
First available in Project Euclid: 6 December 2013

zbMATH: 1305.34152
MathSciNet: MR3161435
Digital Object Identifier: 10.2748/tmj/1386354297

Primary: 34M55
Secondary: 30D35 , 34M05

Keywords: asymptotic solution , growth order , Painlevé equations , value distribution

Rights: Copyright © 2013 Tohoku University


Vol.65 • No. 4 • 2013
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