Tohoku Mathematical Journal
2013
December
SECOND SERIES VOL. 65, NO. 4

Tohoku Math. J.
65 (2013), 591-605

Title FREQUENCY OF $A$-POINTS FOR THE FIFTH AND THE THIRD PAINLEVÉ TRANSCENDENTS IN A SECTOR

Author Shun Shimomura

(Received December 18, 2012)
Abstract. 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.

Mathematics Subject Classification. Primary 34M55; Secondary 34M05, 30D35.
Key words and phrases. Painlevé equations, asymptotic solution, value distribution, growth order.

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