Some evaluation of cubic Euler sums

On harmonic numbers and nonlinear Euler sums

“…The nonlinear Euler sums, i.e., S π,q with π having two or more parts, are more complicated. Such sums were already considered in [3,23,43,46,47,49,51,52]. In [23], Flajolet and Salvy gave an algorithm for reducing S π 1 π 2 ,q to linear Euler sums when π 1 + π 2 + q is even and π 1 , π 2 , q > 1 (see Theorem 4.2 in the reference [23].…”

“…Wang and Lyu [43] shown that all Euler sums of weight eight are reducible to linear sums, and proved that all Euler sums of weight nine are reducible to zeta values. In the most recent papers [44,51,52], the first author has carried out fruitful cooperation with Wang proved that all Euler sums of weight ten can be expressed as a rational linear combination of ζ (10) (−1) j−1 j p and Hn := H(1) n , p ∈ N.…”