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Some evaluation of cubic Euler sums. (English) Zbl 1405.11111
Summary: P. Flajolet and B. Salvy [Exp. Math. 7, No. 1, 15–35 (1998; Zbl 0920.11061)] prove the famous theorem that a nonlinear Euler sum $$S_{i_1 i_2 \cdots i_r, q}$$ reduces to a combination of sums of lower orders whenever the weight $$i_1 + i_2 + \cdots + i_r + q$$ and the order $$r$$ are of the same parity. In this article, we develop an approach to evaluate the cubic sums $$S_{1^2 m, p}$$ and $$S_{1 l_1 l_2, l_3}$$. By using the approach, we establish some relations involving cubic, quadratic and linear Euler sums. Specially, we prove the cubic sums $$S_{1^2 m, m}$$ and $$S_{1(2 l + 1)^2, 2 l + 1}$$ are reducible to zeta values, quadratic and linear sums. Moreover, we prove that the two combined sums involving multiple zeta values of depth four $\mathop{\sum}\limits_{\{i, j \} \in \{1, 2 \}, i \neq j} \zeta(m_i, m_j, 1, 1)\quad \text{and}\quad \mathop{\sum}\limits_{\{i, j, k \} \in \{1, 2, 3 \}, i \neq j \neq k} \zeta(m_i, m_j, m_k, 1)$ can be expressed in terms of multiple zeta values of depth $$\leq 3$$, here $$2 \leq m_1, m_2, m_3 \in \mathbb{N}$$. Finally, we evaluate the alternating cubic Euler sums $$S_{\overline{1}^3, 2 r + 1}$$ and show that it are reducible to alternating quadratic and linear Euler sums. The approach is based on Tornheim type series computations.

MSC:
 11M32 Multiple Dirichlet series and zeta functions and multizeta values
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References:
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