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Diamond lift of Hirose--Sato's formula involving the Hoffman basis

Shin-ichiro Seki

TL;DR

This work proves that Hirose–Sato's explicit Hoffman-basis expansion for a class of multiple zeta values can be derived from the drop 1 relation. By introducing the drop 1 operator $\mathcal{D}$ on the non-commutative algebra $\mathcal{H}$ and establishing the key identity $Z(w_1\tau(w_2))=Z(w_1\star w_2)$ via an induction on weight, the authors provide a new, integer-coefficient proof of the Hirose–Sato expansion. The paper also extends these ideas to the zeta-diamond framework and to finite multiple zeta values, deriving corresponding FMZV corollaries and showing the compatibility with shuffle-type structures. The results connect the drop 1 mechanism with explicit, basis-specific representations of MZVs and offer a pathway to finite-field analogues, strengthening the interplay between algebraic relations and concrete combinatorial expansions.

Abstract

In this paper, we give a new proof of Hirose--Sato's formula for the expansion of $ζ(\{2\}^{a_1-1},3,\dots,\{2\}^{a_r-1},3,\{2\}^{c-1},1,\{2\}^{b_1},\dots, 1,\{2\}^{b_s})$ in the Hoffman basis, using the drop 1 relation.

Diamond lift of Hirose--Sato's formula involving the Hoffman basis

TL;DR

This work proves that Hirose–Sato's explicit Hoffman-basis expansion for a class of multiple zeta values can be derived from the drop 1 relation. By introducing the drop 1 operator on the non-commutative algebra and establishing the key identity via an induction on weight, the authors provide a new, integer-coefficient proof of the Hirose–Sato expansion. The paper also extends these ideas to the zeta-diamond framework and to finite multiple zeta values, deriving corresponding FMZV corollaries and showing the compatibility with shuffle-type structures. The results connect the drop 1 mechanism with explicit, basis-specific representations of MZVs and offer a pathway to finite-field analogues, strengthening the interplay between algebraic relations and concrete combinatorial expansions.

Abstract

In this paper, we give a new proof of Hirose--Sato's formula for the expansion of in the Hoffman basis, using the drop 1 relation.
Paper Structure (4 sections, 7 theorems, 42 equations)

This paper contains 4 sections, 7 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. The drop 1 operator
  3. Proof of Theorem \ref{['thm:main']}
  4. Proof of Corollary \ref{['cor:FMZV']}

Key Result

Theorem 1.1

For positive integers $a, b$, and $c$, we have

Theorems & Definitions (13)

  • Theorem 1.1: Hirose--Sato HiroseSato2019A
  • Theorem 1.2: Hirose--Sato HiroseSato2022+
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1: Hirose--Maesaka--Seki--Watanabe HiroseMaesakaSekiWatanabe2025+
  • Theorem 2.2: Hirose--Maesaka--Seki--Watanabe HiroseMaesakaSekiWatanabe2025+
  • Conjecture 2.3
  • proof : Proof of Theorem $\ref{['thm:main']}$
  • Conjecture 3.1
  • ...and 3 more