Representation of solutions of the Gauss hypergeometric equation by the multiple polylogarithms, functional relations of the multiple polylogarithms and relations of the multiple zeta values

TL;DR

This paper expresses solutions of the Gauss hypergeometric equation using multiple polylogarithms, deriving new relations among these functions and multiple zeta values through connection formulas.

Contribution

It provides a simple series representation of hypergeometric solutions in terms of multiple polylogarithms and explores their functional relations and connections to multiple zeta values.

Findings

Series representation of hypergeometric solutions via multiple polylogarithms

Derivation of new functional relations among multiple polylogarithms

Relations between multiple polylogarithms and multiple zeta values

Abstract

In this article, we express solutions of the Gauss hypergeometric equation as a series of the multiple polylogarithms by using iterated integral. This representation is the most simple case of a semisimple representation of solutions of the formal KZ equation. Moreover, combining this representation with the connection relations of solutions of the Gauss hypergeometric equation, we obtain various relations of the multiple polylogarithms of one variable and the multiple zeta values.