Weighted sum formulas for symmetric multiple zeta values
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).
scientific article; zbMATH DE number 7643077