Multiple polylogarithms, a regularisation process and an admissible open domain of convergence
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| Format: | Preprint |
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2025
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| author | Mehta, Pawan Singh Saha, Biswajyoti |
| author_facet | Mehta, Pawan Singh Saha, Biswajyoti |
| contents | In this article, we study the analytic properties of the multiple polylogarithms in the $s$-aspect. Although the domain of absolute convergence of the series defining the multiple polylogarithms is well-known, the study towards a larger open domain of (conditional) convergence has been limited, particularly when the depth is $\ge 2$. Here, we exhibit a larger open domain of (conditional) convergence for this series by writing certain translation formulas satisfied by them. The series moreover defines a holomorphic function in this open set. We then introduce a regularisation process for the multiple polylogarithms, extending an earlier work of the second author. This regularisation process requires a generalisation of the Euler-Boole summation formula that we derive in the appendix of this article. The regularisation process leads to a larger open domain, where the series (conditionally) converges at integer points. The holomorphicity at such points is a more delicate question and this regularisation process is to be used to study the local behaviour of the multiple polylogarithms around such points. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_00889 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple polylogarithms, a regularisation process and an admissible open domain of convergence Mehta, Pawan Singh Saha, Biswajyoti Number Theory 11M32 In this article, we study the analytic properties of the multiple polylogarithms in the $s$-aspect. Although the domain of absolute convergence of the series defining the multiple polylogarithms is well-known, the study towards a larger open domain of (conditional) convergence has been limited, particularly when the depth is $\ge 2$. Here, we exhibit a larger open domain of (conditional) convergence for this series by writing certain translation formulas satisfied by them. The series moreover defines a holomorphic function in this open set. We then introduce a regularisation process for the multiple polylogarithms, extending an earlier work of the second author. This regularisation process requires a generalisation of the Euler-Boole summation formula that we derive in the appendix of this article. The regularisation process leads to a larger open domain, where the series (conditionally) converges at integer points. The holomorphicity at such points is a more delicate question and this regularisation process is to be used to study the local behaviour of the multiple polylogarithms around such points. |
| title | Multiple polylogarithms, a regularisation process and an admissible open domain of convergence |
| topic | Number Theory 11M32 |
| url | https://arxiv.org/abs/2511.00889 |