### Abstract

For the fifth Painlevé equation near the origin we present two kinds of logarithmic solutions, which are represented, respectively, by convergent series with multipliers admitting asymptotic expansions in descending logarithmic powers and by those with multipliers polynomial in logarithmic powers. It is conjectured that the asymptotic multipliers are also polynomials in logarithmic powers. These solutions are constructed by iteration on certain rings of exponential type series.

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

Pages (from-to) | 797-825 |

Number of pages | 29 |

Journal | Tokyo Journal of Mathematics |

Volume | 39 |

Issue number | 3 |

Publication status | Published - 2017 Mar 1 |

### Fingerprint

### Keywords

- Logarithmic solutions
- Painlevé equation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Tokyo Journal of Mathematics*,

*39*(3), 797-825.

**Logarithmic solutions of the fifth painlevé equation near the origin.** / Shimomura, Shun.

Research output: Contribution to journal › Article

*Tokyo Journal of Mathematics*, vol. 39, no. 3, pp. 797-825.

}

TY - JOUR

T1 - Logarithmic solutions of the fifth painlevé equation near the origin

AU - Shimomura, Shun

PY - 2017/3/1

Y1 - 2017/3/1

N2 - For the fifth Painlevé equation near the origin we present two kinds of logarithmic solutions, which are represented, respectively, by convergent series with multipliers admitting asymptotic expansions in descending logarithmic powers and by those with multipliers polynomial in logarithmic powers. It is conjectured that the asymptotic multipliers are also polynomials in logarithmic powers. These solutions are constructed by iteration on certain rings of exponential type series.

AB - For the fifth Painlevé equation near the origin we present two kinds of logarithmic solutions, which are represented, respectively, by convergent series with multipliers admitting asymptotic expansions in descending logarithmic powers and by those with multipliers polynomial in logarithmic powers. It is conjectured that the asymptotic multipliers are also polynomials in logarithmic powers. These solutions are constructed by iteration on certain rings of exponential type series.

KW - Logarithmic solutions

KW - Painlevé equation

UR - http://www.scopus.com/inward/record.url?scp=85017358511&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017358511&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85017358511

VL - 39

SP - 797

EP - 825

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 3

ER -