Multiple polylogarithms at non-positive indices and combinatorics of Magnus polynomials

Abstract

In this paper we investigate multiple polylogarithms with non-positive multi-indices (nonpositive MPLs) from a combinatorial and algebraic viewpoint. By introducing a correspondence between non-positive multiple polylogarithms and Magnus polynomials in a free associative algebra, we obtain an explicit Magnus-type representation of products of mono-indexed non-positive MPLs. The main identity (Theorem A) expresses such a product as a single non-positive MPL indexed by a Magnus polynomial, which may be regarded as a M\"obius inversion of the expansion formula due to Duchamp-Hoang Ngoc Minh-Ngo. Moreover, we study the effects of permuted indices and show that certain differences of Magnus polynomials belong to the kernel of the linear map ${\rm Li}^-_{\bullet}$ , leading to new functional equations among non-positive MPLs of the same weight and depth. These results clarify the combinatorial structure underlying non-positive MPLs and reveal a close connection with the Magnus expansion in non-commutative algebra.