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Deriving two dualities simultaneously from a family of identities for multiple harmonic sums

Takumi Maesaka, Shin-ichiro Seki, Taiki Watanabe

Abstract

We give a new expression of the multiple harmonic sum, which serves as a refinement of the iterated integral expression of the multiple zeta value, and prove it using the so-called connected sum method. Based on this fact, by taking two kinds of limit operations, we obtain new proofs of both the duality for multiple zeta values and the duality for finite multiple zeta values.

Deriving two dualities simultaneously from a family of identities for multiple harmonic sums

Abstract

We give a new expression of the multiple harmonic sum, which serves as a refinement of the iterated integral expression of the multiple zeta value, and prove it using the so-called connected sum method. Based on this fact, by taking two kinds of limit operations, we obtain new proofs of both the duality for multiple zeta values and the duality for finite multiple zeta values.
Paper Structure (6 sections, 11 theorems, 58 equations)

This paper contains 6 sections, 11 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Deriving the duality in $\mathbb{R}$
  4. Deriving the duality in $\mathcal{A}$
  5. Proof of Theorem \ref{['thm:main']}
  6. Duality in $\widehat{\mathcal{A}}$

Key Result

Theorem 1

Let $k_1,\dots, k_r$ be positive integers with $k_r>1$. Set $k\coloneqq k_1+\cdots+k_r$, $\omega_0(t)\coloneqq\frac{\mathrm{d}t}{t}$, and $\omega_1(t)\coloneqq\frac{\mathrm{d}t}{1-t}$. Then, we have where, for $i\in\{1, k_1+1, k_1+k_2+1, \dots, k_1+\cdots+k_{r-1}+1\}$, $a_i=1$ and for all other $i$, $a_i=0$.

Theorems & Definitions (19)

  • Theorem 1: Iterated integral expression of MZV; widely attributed to Drinfel'd and Kontsevich, and explicitly written in literature by Le--Murakami LeMurakami2005
  • Theorem 1.1: Duality in $\mathbb{R}$
  • Theorem 1.2: Duality in $\mathcal{A}$
  • Theorem 1.3: Discretization of the iterated integral expression of MZV
  • Remark 1.4
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem $\ref{['thm:Rduality']}$ using Theorem $\ref{['thm:main']}$
  • Theorem 3.1: Hoffman Hoffman2015
  • proof : Proof of Theorem $\ref{['thm:nonstarduality']}$ using Theorem $\ref{['thm:main']}$
  • ...and 9 more