## 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 |

DOIs | |

Publication status | Published - 2004 Jul |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)