### Abstract

In this article, we show that the generalized inversion formulas of the multiple polylogarithms of one variable, which are generalizations of the inversion formula of the dilogarithm, characterize uniquely the multiple polylogarithms under certain conditions. This means that the multiple polylogarithms are constructed from the multiple zeta values. We call such a problem of determining certain functions a recursive Riemann-Hilbert problem of additive type. Furthermore we show that the fundamental solutions of the KZ equation of one variable are uniquely characterized by the connection relation between the fundamental solutions of the KZ equation normalized at z = 0 and z = 1 under some assumptions. Namely the fundamental solutions of the KZ equation are constructed from the Drinfel'd associator. We call this problem a Riemann-Hilbert problem of multiplicative type.

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

Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Tokyo Journal of Mathematics |

Volume | 41 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2018 Jun 1 |

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### Keywords

- KZ equation
- Multiple polylogarithms
- Multiple zeta values

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Fundamental solutions of the knizhnik-zamolodchikov equation of one variable and the riemann-hilbert problem.** / Oi, Shu; Ueno, Kimio.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Fundamental solutions of the knizhnik-zamolodchikov equation of one variable and the riemann-hilbert problem

AU - Oi, Shu

AU - Ueno, Kimio

PY - 2018/6/1

Y1 - 2018/6/1

N2 - In this article, we show that the generalized inversion formulas of the multiple polylogarithms of one variable, which are generalizations of the inversion formula of the dilogarithm, characterize uniquely the multiple polylogarithms under certain conditions. This means that the multiple polylogarithms are constructed from the multiple zeta values. We call such a problem of determining certain functions a recursive Riemann-Hilbert problem of additive type. Furthermore we show that the fundamental solutions of the KZ equation of one variable are uniquely characterized by the connection relation between the fundamental solutions of the KZ equation normalized at z = 0 and z = 1 under some assumptions. Namely the fundamental solutions of the KZ equation are constructed from the Drinfel'd associator. We call this problem a Riemann-Hilbert problem of multiplicative type.

AB - In this article, we show that the generalized inversion formulas of the multiple polylogarithms of one variable, which are generalizations of the inversion formula of the dilogarithm, characterize uniquely the multiple polylogarithms under certain conditions. This means that the multiple polylogarithms are constructed from the multiple zeta values. We call such a problem of determining certain functions a recursive Riemann-Hilbert problem of additive type. Furthermore we show that the fundamental solutions of the KZ equation of one variable are uniquely characterized by the connection relation between the fundamental solutions of the KZ equation normalized at z = 0 and z = 1 under some assumptions. Namely the fundamental solutions of the KZ equation are constructed from the Drinfel'd associator. We call this problem a Riemann-Hilbert problem of multiplicative type.

KW - KZ equation

KW - Multiple polylogarithms

KW - Multiple zeta values

UR - http://www.scopus.com/inward/record.url?scp=85051164074&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85051164074&partnerID=8YFLogxK

U2 - 10.3836/tjm/1502179274

DO - 10.3836/tjm/1502179274

M3 - Article

VL - 41

SP - 1

EP - 20

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 1

ER -