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Evaluations of some quadruple Euler sums of even weight. (English) Zbl 1243.40004
The authors obtain evaluations of some quadruple Euler sums of even weight by the identities among multiple zeta values with variables and relations obtained from the shuffle formula of two multiple zeta values.
Authors’ abstract: For positive integers $$\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r}$$ with $$\alpha_{r} \geq 2$$, the multiple zeta value or $$r$$-fold Euler sum is defined by $\zeta(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r}) = \sum_{1\leq n_{1}<n_{2}<\cdots<n_{r}}n_1^{-\alpha_1}n_2^{-\alpha_2}\cdots n_r^{-\alpha_r}$ where $$|\alpha|=\alpha_1+\alpha_2+\cdots+\alpha_r$$ and $$r$$ are the weight and depth of $$\zeta(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r})$$ respectively. By the general theorem given in [C. Markett, J. Number Theory 48, 113–132 (1994; Zbl 0810.11047)]], the multiple zeta value $$\zeta(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r})$$ can be expressed as a rational linear combination of products of multiple zeta values of lower depth if its depth and weight are of different parity. In other words, when the sum of its depth and weight is odd. However, there are still some exceptions for quadruple Euler sums. As conjectured in [J. M. Borwein et al., Trans. Am. Math. Soc. 353, 907–941 (2001; Zbl 1002.11093)], a quadruple Euler sum with even weight exceeding 14 can be expressed as a rational linear combination of products of multiple zeta values of depth 1, 2 and 3 if and only if it is one of the following forms: $$\zeta(1,a,b,a)$$, $$\zeta(b,1,a,a)$$, $$\zeta(b,b,1,a)$$ or $$\zeta(a,b,b,a)$$ with $$a=b$$ or $$b=1$$. In this paper, we evaluate these quadruple Euler sums of even weight by the identities among multiple zeta values with variables and relation obtained from the shuffle formula of two multiple zeta values.

##### MSC:
 40A25 Approximation to limiting values (summation of series, etc.) 40B05 Multiple sequences and series 11M32 Multiple Dirichlet series and zeta functions and multizeta values 11M35 Hurwitz and Lerch zeta functions
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