Explicit formulas of Euler sums via multiple zeta values

Abstract

Flajolet and Salvy pointed out that every Euler sum is a Q-linear combination of multiple zeta values. However, in the literature, there is no formula completely revealing this relation. In this paper, using permutations and compositions, we establish two explicit formulas for the Euler sums, and show that all the Euler sums are indeed expressible in terms of MZVs. Moreover, we apply this method to the alternating Euler sums, and show that all the alternating Euler sums are reducible to alternating MZVs. Some famous results, such as the Euler theorem, the Borwein-Borwein-Girgensohn theorems, and the Flajolet-Salvy theorems can be obtained directly from our theory. Some other special cases, such as the explicit expressions of S r m ,q , Srm ,q , S r m ,q and Srm ,q , are also presented here. The corresponding Maple programs are developed to help us compute all the sums of weight w ≤ 11 for non-alternating case and of weight w ≤ 6 for alternating case.

Extensions of Euler-Type Sums and Ramanujan-Type Sums

“…The study of nonlinear Euler sums has attracted a lot of research in the last three decades. Some related results may be seen in the works of [16,21,22,23,25] and the references therein.…”

“…It should be emphasized that every (alternating) Euler sum of weight w and degree n is clearly a Q-linear combination of (alternating) multiple zeta values of weight w and depth less than or equal to n + 1 (an explicit formula was given by the author and Wang [23]). The multiple zeta values are defined by [13,24,25]…”

“…Rational relations among alternating multiple zeta values are also tabulated in the MZVDM (through weight 12 [8]). With the help of results of [8], the author and Wang [23] have developed a Maple package to evaluate the non-alternating Euler sums of weight 2 ≤ w ≤ 16 and the alternating Euler sums of weight 1 ≤ w ≤ 6.…”

“…For example, Zhao [26] gave explicit evaluations of all 89 alternating Euler sums with weight ≤ 5. The author and Wang [23] have developed a Maple package to evaluate the non-alternating Euler sums of weight 2 ≤ w ≤ 16 and the alternating Euler sums of weight 1 ≤ w ≤ 6.…”

On variants of the Euler sums and symmetric extensions of the Kaneko-Tsumura conjecture

“…≤ p k and q ≥ 2, and the linear sums are of the form S p,q . For an early introduction and study on the evaluations of the classical Euler sums, the readers may consult in Flajolet and Salvy's paper [6], and for some recent progress, the readers are referred to [20,23,25] and references therein. In this paper, using various expansions of the parametric digamma function and the method of residue computations, we establish symmetric extensions of the Kaneko-Tsumura conjecture (1.2) on three variants of the linear Euler sums, defined by…”