2025/12/18 by Kitamura, Kohei
Mathematics · #05A19 (Secondary) #11G55 (Primary) 16S30 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2512.16195
openalex publication_date 2025/12/18 · openalex created_date 2025/12/21 · openalex updated_date 2026/07/28
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 Möbius 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.