Iterated integrals associated with colored rooted trees

TL;DR

This paper introduces a new class of iterated integrals linked to colored rooted trees and proves shuffle relations for certain p-adic and t-adic symmetric polylogarithms, extending previous work on finite multiple zeta values.

Contribution

It generalizes the theory of finite multiple zeta values to include iterated integrals associated with colored rooted trees, providing new proofs of shuffle relations.

Findings

Established shuffle relations for p-adic finite and t-adic symmetric polylogarithms.

Extended the framework of finite multiple zeta values to colored rooted trees.

Provided a new approach to understanding polylogarithms through combinatorial structures.

Abstract

In this paper, we introduce iterated integrals associated with colored rooted trees and give proofs for the shuffle relations for $\boldsymbol{p}$-adic finite and $t$-adic symmetric polylogarithms. This method generalizes the theory of the finite multiple zeta values associated with $2$-colored rooted trees introduced by Ono.