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

Shu Oi, Kimio Ueno

Research output: Contribution to journalArticle

### 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 1-20 20 Tokyo Journal of Mathematics 41 1 https://doi.org/10.3836/tjm/1502179274 Published - 2018 Jun 1

### Fingerprint

Polylogarithms
Riemann-Hilbert Problem
Fundamental Solution
Inversion Formula
Dilogarithm
Multiple zeta Values
Multiplicative

### Keywords

• KZ equation
• Multiple polylogarithms
• Multiple zeta values

### ASJC Scopus subject areas

• Mathematics(all)

### Cite this

In: Tokyo Journal of Mathematics, Vol. 41, No. 1, 01.06.2018, p. 1-20.

Research output: Contribution to journalArticle

title = "Fundamental solutions of the knizhnik-zamolodchikov equation of one variable and the riemann-hilbert problem",
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.",
keywords = "KZ equation, Multiple polylogarithms, Multiple zeta values",
author = "Shu Oi and Kimio Ueno",
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pages = "1--20",
journal = "Tokyo Journal of Mathematics",
issn = "0387-3870",
publisher = "Publication Committee for the Tokyo Journal of Mathematics",
number = "1",

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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

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