### Abstract

The first Painlevé hierarchy, which is a sequence of higher order analogues of the first Painlevé equation, follows from the singular manifold equations for the mKdV hierarchy. For meromorphic solutions of the first Painlevé hierarchy, we give a lower estimate for the number of poles; which is regarded as an extension of one corresponding to the first Painlevé equation, and which indicates a conjecture on the growth order. From our main result, two corollaries follow: one is the transcendency of meromorphic solutions, and the other is a lower estimate for the frequency of α-points. An essential part of our proof is estimation of certain sums concerning the poles of each meromorphic solution.

Original language | English |
---|---|

Pages (from-to) | 471-485 |

Number of pages | 15 |

Journal | Publications of the Research Institute for Mathematical Sciences |

Volume | 40 |

Issue number | 2 |

Publication status | Published - 2004 Jul |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Publications of the Research Institute for Mathematical Sciences*,

*40*(2), 471-485.

**Poles and α-points of meromorphic solutions of the first Painlevé hierarchy.** / Shimomura, Shun.

Research output: Contribution to journal › Article

*Publications of the Research Institute for Mathematical Sciences*, vol. 40, no. 2, pp. 471-485.

}

TY - JOUR

T1 - Poles and α-points of meromorphic solutions of the first Painlevé hierarchy

AU - Shimomura, Shun

PY - 2004/7

Y1 - 2004/7

N2 - The first Painlevé hierarchy, which is a sequence of higher order analogues of the first Painlevé equation, follows from the singular manifold equations for the mKdV hierarchy. For meromorphic solutions of the first Painlevé hierarchy, we give a lower estimate for the number of poles; which is regarded as an extension of one corresponding to the first Painlevé equation, and which indicates a conjecture on the growth order. From our main result, two corollaries follow: one is the transcendency of meromorphic solutions, and the other is a lower estimate for the frequency of α-points. An essential part of our proof is estimation of certain sums concerning the poles of each meromorphic solution.

AB - The first Painlevé hierarchy, which is a sequence of higher order analogues of the first Painlevé equation, follows from the singular manifold equations for the mKdV hierarchy. For meromorphic solutions of the first Painlevé hierarchy, we give a lower estimate for the number of poles; which is regarded as an extension of one corresponding to the first Painlevé equation, and which indicates a conjecture on the growth order. From our main result, two corollaries follow: one is the transcendency of meromorphic solutions, and the other is a lower estimate for the frequency of α-points. An essential part of our proof is estimation of certain sums concerning the poles of each meromorphic solution.

UR - http://www.scopus.com/inward/record.url?scp=4544343746&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4544343746&partnerID=8YFLogxK

M3 - Article

VL - 40

SP - 471

EP - 485

JO - Publications of the Research Institute for Mathematical Sciences

JF - Publications of the Research Institute for Mathematical Sciences

SN - 0034-5318

IS - 2

ER -