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

Shun Shimomura

Research output: Contribution to journalArticle

8 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
Publication statusPublished - 2004 Jul

Fingerprint

Meromorphic Solution
Pole
Estimate
Corollary
Higher Order
Analogue
Hierarchy

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Publications of the Research Institute for Mathematical Sciences, Vol. 40, No. 2, 07.2004, p. 471-485.

Research output: Contribution to journalArticle

@article{7e0e85b99a6742fab15cb51007dcfb0d,
title = "Poles and α-points of meromorphic solutions of the first Painlev{\'e} hierarchy",
abstract = "The first Painlev{\'e} hierarchy, which is a sequence of higher order analogues of the first Painlev{\'e} equation, follows from the singular manifold equations for the mKdV hierarchy. For meromorphic solutions of the first Painlev{\'e} hierarchy, we give a lower estimate for the number of poles; which is regarded as an extension of one corresponding to the first Painlev{\'e} 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.",
author = "Shun Shimomura",
year = "2004",
month = "7",
language = "English",
volume = "40",
pages = "471--485",
journal = "Publications of the Research Institute for Mathematical Sciences",
issn = "0034-5318",
publisher = "European Mathematical Society Publishing House",
number = "2",

}

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 -