Some evaluation of cubic Euler sums

Abstract

P. Flajolet and B. Salvy [15] prove the famous theorem that a nonlinear Euler sum S i 1 i 2 ···ir,q reduces to a combination of sums of lower orders whenever the weight i 1 + i 2 + · · · + 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 1l 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(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 fourcan be expressed in terms of multiple zeta values of depth ≤ 3, here 2 ≤ m 1 , m 2 , m 3 ∈ N. Finally, we evaluate the alternating cubic Euler sums S13 ,2r+1 and show that it are reducible to alternating quadratic and linear Euler sums. The approach is based on Tornheim type series computations.

Explicit formulas of Euler sums via multiple zeta values

“…Just as the non-alternating case, using the function Using the tables of alternating MZVs given in [5], we have computed all the alternating Euler sums of weight w ≤ 6. The evaluations of some alternating Euler sums of weight w ≤ 5 can be found in [44,[47][48][49]54], so we list in the following table, as examples, the evaluations of 18 alternating Euler sums of weight 6, which are of the form…”

“…Recently, rapid progress has been made in this field. Using the Bell polynomials, generating functions, integrals of special functions, multiple zeta (star) values, the Stirling sums and the Tornheim type series, we study the (alternating) Euler sums systematically [42][43][44][46][47][48][49][50][51]53,54]. As a consequence, the evaluation of all the unknown Euler sums up to the weight 11 are presented, and a basis of Euler sums of weight 3 ≤ w ≤ 11 is…”

“…Using the tables of alternating MZVs given in [5], we have computed all the alternating Euler sums of weight w ≤ 6. The evaluations of some alternating Euler sums of weight w ≤ 5 can be found in [44,[47][48][49]54], so we list in the following…”

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) , ζ 2 (5) , ζ where the alternating Riemann zeta function is defined by (for more details, see [23])…”