Evaluating Generating Functions for Periodic Multiple Polylogarithms

TL;DR

This paper introduces a systematic numerical evaluation method for generating functions of periodic multiple polylogarithms using Chen-Fliess series, enabling validation of conjectures and extending to non-periodic cases.

Contribution

It presents a novel approach to evaluate multiple polylogarithm generating functions via bilinear dynamical systems and generalizes to non-periodic components.

Findings

Numerical validation of the Hoffman conjecture.

Development of a bilinear dynamical system realization.

Extension to non-periodic multiple polylogarithms.

Abstract

The goal of the paper is to give a systematic way to numerically evaluate the generating function of a periodic multiple polylogarithm using a Chen-Fliess series with a rational generating series. The idea is to realize the corresponding Chen-Fliess series as a bilinear dynamical system. A standard form for such a realization is given. The method is also generalized to the case where the multiple polylogarithm has non-periodic components. This allows one, for instance, to numerically validate the Hoffman conjecture. Finally, a setting in terms of dendriform algebras is provided.