Multiple polylogarithms at non-positive indices and combinatorics of Magnus polynomials
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| Format: | Preprint |
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2025
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| _version_ | 1866912813619871744 |
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| author | Kitamura, Kohei |
| author_facet | Kitamura, Kohei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_16195 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple polylogarithms at non-positive indices and combinatorics of Magnus polynomials Kitamura, Kohei Combinatorics Number Theory 11G55 (Primary) 16S30, 05A19 (Secondary) 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. |
| title | Multiple polylogarithms at non-positive indices and combinatorics of Magnus polynomials |
| topic | Combinatorics Number Theory 11G55 (Primary) 16S30, 05A19 (Secondary) |
| url | https://arxiv.org/abs/2512.16195 |