Quasi-derivation relations for multiple zeta values revisited

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Quasi‑derivation relations for multiple zeta values revisited Masanobu Kaneko1 · Hideki Murahara2 · Takuya Murakami3 Received: 21 July 2019 / Accepted: 22 October 2020 © The Author(s), under exclusive licence to Mathematisches Seminar der Universität Hamburg 2020

Abstract We take another look at the so-called quasi-derivation relations in the theory of multiple zeta values, by giving a certain formula for the quasi-derivation operator. In doing so, we are not only able to prove the quasi-derivation relations in a simpler manner but also give an analog of the quasi-derivation relations for finite multiple zeta values. Keywords  Multiple zeta values · Finite multiple zeta values · Derivation relations · Quasiderivation relations Mathematics Subject Classification  Primary 11M32 · Secondary 05A19

1 Introduction The quasi-derivation relations in the theory of multiple zeta values is a generalization, proposed by the first-named author and established by T. Tanaka, of a set of linear relations known as derivation relations, which we are first going to recall. We use Hoffman’s algebraic setup ([5]) with a slightly different convention. Let ℌ ∶= ℚ⟨x, y⟩ be the noncommutative polynomial algebra in two indeterminates x and y. This was introduced in order to encode multiple zeta values in the way the monomial yxk1 −1 yxk2 −1 … yxkr −1 corresponds to the multiple zeta value

Communicated by Herr Kühn. * Hideki Murahara hmurahara@nakamura‑u.ac.jp Masanobu Kaneko [email protected]‑u.ac.jp Takuya Murakami [email protected] 1

Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi‑ku, Fukuoka 819‑0395, Japan

2

Nakamura Gakuen University Graduate School, 5‑7‑1, Befu, Jonan‑ku, Fukuoka 814‑0198, Japan

3

Graduate School of Mathematics, Kyushu University, 744, Motooka, Nishi‑ku, Fukuoka 819‑0395, Japan



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𝜁 (k1 , k2 , … , kr ) ∶=



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k1 k2 0