Weighted sum formulas for symmetric multiple zeta values
type: Item
instance of: scholarly article
title: Weighted sum formulas for symmetric multiple zeta values
author: Kento Fujita
author: Yasushi Komori
published in: The Ramanujan Journal
publication date: 2023-01-17T00:00:00Z
review text: Let \(\mathbf{k}=(k_1,\ldots,k_r)\in \mathbb{N}^r\) and \(k_r>1\). The multiple zeta value (MZV) is the real number defined by \[ \zeta(\mathbf{k})=\sum_{0<m_1<\cdots <m_r} \frac{1}{m_1^{k_1}\ldots m_r^{k_r}}. \] Let \(\mathcal{Z}\) be the \(\mathbb{Q}\)-algebra of generated by all multiple zeta values (MZVs). Then, the symmetric multiple zeta values (SMZVs) are elements of \(\mathcal{Z}/\zeta(2)\mathcal{Z}\) and were introduced by M. Kaneko and D. Zagier as real counterparts of finite multiple zeta values (FMZVs). It is well-known that symmetric multiple zeta values (SMZVs) satisfy double shuffle relations and duality relations. In the paper [Doc. Math. 25, 365--380 (2020; Zbl 1448.11165)], \textit{M. Hirose} introduced certain lifts of SMZVs which live in the ring generated by all multiple zeta values and \(2\pi i\) as certain iterated integrals on \(\mathbb{P}_1-\lbrace 0,1,\infty \rbrace\) along a certain closed path. These lifted values are called refined symmetric multiple zeta values (RSMZVs). In the paper under review, the author derives sum formulas with four parameters among symmetric multiple zeta values (SMZVs) by iterated integral expressions of refined symmetric multiple zeta values (RSMZVs).
reviewed by: Sami Omar
Mathematics Subject Classification ID: 11M32
zbMATH DE Number: 7643077
zbMATH Keywords: symmetric multiple zeta value
zbMATH Keywords: refined symmetric multiple zeta value
MaRDI profile type: Publication
full work available at URL: s11139-022-00656-3
OpenAlex ID: W4307130837
cites work: Certain weighted sum formulas for multiple zeta values with some parameters
cites work: Weighted sum formulas for finite multiple zeta values
cites work: Derivation and double shuffle relations for multiple zeta values
cites work: Double shuffle relations for refined symmetric multiple zeta values
cites work: A note on finite real multiple zeta values
DOI: 10.1007/S11139-022-00656-3
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description: scientific article; zbMATH DE number 7643077