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Multiple polylogarithms, a regularisation process and an admissible open domain of convergence

Pawan Singh Mehta, Biswajyoti Saha

TL;DR

This work analyzes the analytic structure of multiple polylogarithms in the $s$-aspect, expanding the domain of convergence beyond the classical region by translating partial tails of the defining series and establishing holomorphy on an enlarged open set $U_r(\mathbf{z})$. It then develops a robust regularisation procedure for special values at roots of unity, grounded in asymptotic expansions relative to a comparison scale $\mathcal{E}$ and a finite-dimensional algebra $\mathcal{C}$, including a generalized Euler–Boole summation formula. The authors prove that the multiple Dirichlet series converges on $U_r(\mathbf{z})$ and that integral points in the larger domain $V_r(\mathbf{z})$ admit a regularised, well-defined value, with explicit asymptotic expansions and Stieltjes-constant-type coefficients. Collectively, these results provide a deeper analytic framework for the local behavior of multiple polylogarithms near integral points and lay groundwork for further study of their regularised values and holomorphic properties across extended domains.

Abstract

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.

Multiple polylogarithms, a regularisation process and an admissible open domain of convergence

TL;DR

This work analyzes the analytic structure of multiple polylogarithms in the -aspect, expanding the domain of convergence beyond the classical region by translating partial tails of the defining series and establishing holomorphy on an enlarged open set . It then develops a robust regularisation procedure for special values at roots of unity, grounded in asymptotic expansions relative to a comparison scale and a finite-dimensional algebra , including a generalized Euler–Boole summation formula. The authors prove that the multiple Dirichlet series converges on and that integral points in the larger domain admit a regularised, well-defined value, with explicit asymptotic expansions and Stieltjes-constant-type coefficients. Collectively, these results provide a deeper analytic framework for the local behavior of multiple polylogarithms near integral points and lay groundwork for further study of their regularised values and holomorphic properties across extended domains.

Abstract

In this article, we study the analytic properties of the multiple polylogarithms in the -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 . 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.

Paper Structure

This paper contains 5 sections, 12 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Regularisation process for the special values of the multiple polylogarithms and the higher order derivatives
  3. Proof of Theorem \ref{['big-dom']}
  4. Proof of Theorem \ref{['ad-dom']}
  5. Generalised Euler-Boole summation formula

Key Result

Theorem 1

The function $\mathrm{Li}_{(z_1,\ldots,z_r)}(s_1,\ldots,s_r)$ extends to a meromorphic function on the whole of ${\mathbb C}^r$. If $z_{[1,i]} \neq 1$ for all $1 \le i \le r$, then $\mathrm{Li}_{(z_1,\ldots,z_r)}(s_1,\ldots,s_r)$ is holomorphic on ${\mathbb C}^r$. Otherwise, if $i_1 < \cdots < i_m$ $(b)$ If $i_1 \not =1$, then $\mathrm{Li}_{(z_1,\ldots,z_r)}(s_1,\ldots,s_r)$ is holomorphic outsid

Theorems & Definitions (30)

  • Theorem 1: Saha
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Definition 2
  • Example 1
  • Proposition 1
  • proof
  • Theorem 4
  • proof
  • ...and 20 more