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

Shun Shimomura

Research output: Contribution to journalArticle

9 Citations (Scopus)

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 languageEnglish
Pages (from-to)471-485
Number of pages15
JournalPublications of the Research Institute for Mathematical Sciences
Volume40
Issue number2
DOIs
Publication statusPublished - 2004 Jul

ASJC Scopus subject areas

  • Mathematics(all)

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