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Some identities for multiple alternating zeta values. (English) Zbl 1483.11189

The sum of all the multiple zeta values (MZVs) of weight \(mn\) and depth \(k\) with the MZV arguments being multiples of a fixed integer \(m\ge2\) are denoted in the present paper by \(A(m,n,k)\). In formula, \[ A(m, n, k)=\sum_{\substack{s_{1}+s_{2}+\cdots+s_{k}=n \\ s_{j} \in \mathbb{N}}} \zeta\left(m \overline{s_{1}}, m \overline{s_{2}}, \cdots, m \overline{s_{k}}\right). \] One main result of the paper is an explicit expression for \(A(m, n, k)\) in terms of alternating zeta and zeta star values (MZSVs): \[ A(m, n, k)=\sum_{\substack{a+b=n \\ a, b \in \mathbb{N}_{0}}}(-1)^{a-k}\left(\begin{array}{l}a\\ k\end{array}\right) \cdot \zeta\left(\{\bar{m}\}^{a}\right) \cdot \zeta^{\star}\left(\{\bar{m}\}^{b}\right). \] (Here the standard notations of MZV theory apply.)
Another result is a finite expression for alternating MZVs and MZSVs with repeated argument. The simpler of these reads as \[ \zeta^*\left(\{\overline{2 m}\}^{n}\right)= \] \[ \frac{\pi^{2 m n}}{2^{2 m n}} \sum_{\sum_{j=1}^{m} n_{j}=m n} \prod_{j=1}^m \prod_{k_{j}+l_{j}=n_{j}} \frac{(-1)^{n_{j}-1}\left(2^{2 k_{j}}-2\right) B_{2 k_{j}} E_{2 l_{j}} \mathrm{e}^{\frac{\pi i}{m}\left(2 j n_{j}-l_{j}\right)}}{\left(2 k_{j}\right) !\left(2 l_{j}\right) !}. \] This result leads to a simple evaluation for \(A(2,n,k)\) (see Corollary 1 in the paper).

MSC:

11M32 Multiple Dirichlet series and zeta functions and multizeta values
11M35 Hurwitz and Lerch zeta functions
11B68 Bernoulli and Euler numbers and polynomials
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