Shuffle products for multiple zeta values and partial fraction decompositions of zeta-functions of root systems

TL;DR

This paper reveals a new interpretation of the shuffle product for multiple zeta values, linking it to partial fraction decompositions of zeta-functions of root systems, and provides alternative proofs of double shuffle relations.

Contribution

It introduces a novel perspective connecting shuffle products with partial fraction decompositions, simplifying proofs of double shuffle relations without integral expressions.

Findings

Shuffle product coincides with partial fraction decompositions.

Extended double shuffle relations are proved without Drinfel'd integrals.

Functional relations including double shuffle relations are derived.

Abstract

The shuffle product plays an important role in the study of multiple zeta values. This is expressed in terms of multiple integrals, and also as a product in a certain non-commutative polynomial algebra over the rationals in two indeterminates. In this paper, we give a new interpretation of the shuffle product. In fact, we prove that the procedure of shuffle products essentially coincides with that of partial fraction decompositions of multiple zeta values of root systems. As an application, we give a proof of extended double shuffle relations without using Drinfel'd integral expressions for multiple zeta values. Furthermore, our argument enables us to give some functional relations which include double shuffle relations.