Some evaluation of cubic Euler sums

P. Flajolet and B. Salvy [15] prove the famous theorem that a nonlinear Euler sum Si1i2⋯ir,q reduces to a combination of sums of lower orders whenever the weight i1+i2+⋯+ir+q and the order r are of the same parity. In this article, we develop an approach to evaluate the cubic sums S12m,p and S1l1l2,l3. By using the approach, we establish some relations involving cubic, quadratic and linear Euler sums. Specially, we prove the cubic sums S12m,m and S1(2l+1)2,2l+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∑{i,j}∈{1,2},i≠jζ(mi,mj,1,1)and∑{i,j,k}∈{1,2,3},i≠j≠kζ(mi,mj,mk,1) can be expressed in terms of multiple zeta values of depth ≤3, here 2≤m1,m2,m3∈N. Finally, we evaluate the alternating cubic Euler sums S1¯3,2r+1 and show that it are reducible to alternating quadratic and linear Euler sums. The approach is based on Tornheim type series computations.