Depth reductions for associators

TL;DR

This paper establishes depth reduction formulas for associators, linking their coefficients to lower-depth coefficients, and explores implications for $p$-adic and finite multiple zeta values through explicit formulas.

Contribution

It introduces new depth reduction formulas for associator coefficients and applies these to study $p$-adic and finite multiple zeta values.

Findings

Depth reduction formulas for associator coefficients.

New perspectives on relating $p$-adic and finite multiple zeta values.

Explicit formulas connecting associators to $p$-adic multiple zeta values.

Abstract

We prove that for any associator, two specific families of coefficients of the associator can be expressed in terms of coefficients of lower depth. Combining these results to our notions of adjoint $p$-adic multiple zeta values and multiple harmonic values, we obtain a new point of view on the question of relating $p$-adic and finite multiple zeta values, and a few other application to the study of $p$-adic multiple zeta values via explicit formulas.