The algebra and combinatorics of shuffles and multiple zeta values
The authors consider multiple zeta values \[ \zeta (s_1,\ldots ,s_k)=\sum_{n_1>n_2>\ldots >n_k>0} \prod_{j=1}^kn_j^{-s_j} \] where \(s_1,\ldots ,s_k\) are positive integers, \(s_1>1\). In particular, given a vector \(\vec s=(m_0,m_1,\ldots ,m_{2n})\) of non-negative integers, they study \[ Z(\vec s)=\zeta \left( \{2\}^{m_0},3,\{2\}^{m_1},1,\{2\}^{m_2},3,\{2\}^{m_3},1,\ldots ,3,\{2\}^{m_{2n-1}},1,\{2\}^{m_{2n}}\right) \] where \(\{2\}^m\) means the argument 2 repeated \(m\) times. Sums of \(Z(\vec s)\) over some sets of \(\vec s\) are found explicitly, generalizing the formula for \(\zeta (\{ 3,1\}^n)\) conjectured by D. Zagier and proved by \textit{J. M. Borwein, D. M. Bradley, D. J. Broadhurst}, and \textit{P. Lisonek} [Trans. Am. Math. Soc. 353, 907-941 (2001; Zbl 1002.11093)]. The technique is based on further development of the algebraic and combinatorial theory of shuffles introduced by \textit{K.-T. Chen} [Proc. Lond. Math. Soc. (3) 4, 502-512 (1954; Zbl 0058.25603)] and \textit{R. Ree} [Ann. Math. (2) 68, 210-220 (1958; Zbl 0083.25401)].
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