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**Frequency of \(a\)-points for the fifth and the third Painlevé transcendents in a sector.**
*(English)*
Zbl 1305.34152

In this paper value distribution properties for solutions of the fifth and the third Painlevé equations in a sector are studied. Supposing that the values taken along some curve tending to infinity are bounded away from 1 and another specified complex number, the author obtains new good upper estimates, which are expected to be the best possible for general solutions, for the number of \(a\)-points (i.e., points where a solution in question takes the value \(a\); \(a\) is allowed to be infinity, that is, the number of poles are also included) and for the growth order in a sector containing the curve. The main results are applicable to almost all known asymptotic solutions of the fifth Painlevé equation: Using the main results, the author discusses also the frequency of \(a\)-points, the equi-distribution property, and the growth order for several concrete asymptotic solutions as well.

Reviewer: Yoshitsugu Takei (Kyoto)

### MSC:

34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |

34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |

30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |

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\textit{S. Shimomura}, Tôhoku Math. J. (2) 65, No. 4, 591--605 (2013; Zbl 1305.34152)

### References:

[1] | F. V. Andreev and A. V. Kitaev, Exponentially small corrections to divergent asymptotic expansions of solutions of the fifth Painlevé equation, Math. Res. Lett. 4 (1997), 741-759. · Zbl 0889.34003 |

[2] | F. V. Andreev and A. V. Kitaev, Connection formulae for asymptotics of the fifth Painlevé transcendent on the real axis, Nonlinearity 13 (2000), 1801-1840. · Zbl 0970.34076 |

[3] | G. Barsegian, I. Laine and D. T. Lê, On topological behaviour of solutions of some algebraic differential equations, Complex Var. Elliptic Equ. 53 (2008), 411-421. · Zbl 1163.34057 |

[4] | A. A. Goldberg and I. V. Ostrovskii, Value distribution of meromorphic functions, Translations of Mathematical Monographs 236, American Mathematical Society, Providence, 2008. · Zbl 1152.30026 |

[5] | B. Ja. Levin and I. V. Ostrovskii, On the dependence of the growth of an entire function on the distribution of the zeros of its derivatives (in Russian), Sibirsk. Mat. Zh. 1 (1960), 427-455; English translation in: Amer. Math. Soc. Transl. 32 (1963), 323-357. |

[6] | Y. Sasaki, Value distribution of the fifth Painlevé transcendents in sectorial domains, J. Math. Anal. Appl. 330 (2007), 817-828. · Zbl 1133.34049 |

[7] | Y. Sasaki, Value distribution of the third Painlevé transcendents in sectorial domains, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), 79-82. · Zbl 1152.34396 |

[8] | S. Shimomura, Analytic integration of some nonlinear ordinary differential equations and the fifth Painlevé equation in the neighbourhood of an irregular singular point, Funkcial. Ekvac. 26 (1983), 301-338. · Zbl 0548.34004 |

[9] | S. Shimomura, On solutions of the fifth Painlevé equation on the positive real axis. II, Funkcial. Ekvac. 30 (1987), 203-224. · Zbl 0654.34049 |

[10] | S. Shimomura, Proofs of the Painlevé property for all Painlevé equations, Japan. J. Math. 29 (2003), 159-180. · Zbl 1085.34071 |

[11] | S. Shimomura, Growth of modified Painlevé transcendents of the fifth and the third kind, Forum Math. 16 (2004), 231-247. · Zbl 1058.34127 |

[12] | S. Shimomura, Equi-distribution of values for the third and the fifth Painlevé transcendents, Nagoya Math. J. 192 (2008), 89-109. · Zbl 1163.34062 |

[13] | S. Shimomura, Truncated solutions of the fifth Painlevé equation, Funkcial. Ekvac. 54 (2011), 451-471. · Zbl 1235.34236 |

[14] | K. Takano, A 2-parameter family of solutions of Painlevé equation (V) near the point at infinity, Funkcial. Ekvac. 26 (1983), 79-113. · Zbl 0528.34005 |

[15] | M. Tsuji, On Borel’s directions of meromorphic functions of finite order, Tohoku Math. J. (2) (1950), 97-112. · Zbl 0041.20202 |

[16] | S. Wang, On the sectorial oscillation theory of \(f''+A(z)f=0\), Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 92 (1994). · Zbl 0824.30020 |

[17] | S. Yoshida, 2-parameter family of solutions for Painlevé equations (I)-(V) at an irregular singular point, Funkcial. Ekvac. 28 (1985), 233-248. · Zbl 0592.34036 |

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