Restricted sum formula of alternating Euler sums

Abstract

In this paper we study restricted sum formulas involving alternating Euler sums which are defined byfor all positive integers s 1 , . . . , s d andWe call w = s 1 + · · · + s d the weight and d the depth. When ε j = −1 we say the jth component is alternating. We first consider Euler sums of the following special type:For d ≤ n, let Ξ(2n, d) be the sum of all ξ(2s 1 , . . . , 2s d ) of fixed weight 2n and depth d. We derive a formula for Ξ(2n, d) using the theory of symmetric functions established by Hoffman recently. We also consider restricted sum formulas of Euler sums with fixed weight 2n, depth d and fixed number α of alternating components at even arguments. When α = 1 or α = d we can determine precisely the restricted sum formulas. For other α we only treat the cases d < 5 completely since the symmetric function theory becomes more and more unwieldy to work with when α moves closer to d/2.

Restricted sum formula of multiple zeta values

“…Recently, Hoffman [3] extended (2) to arbitrary depths. Moreover, similar formulas have been obtained for some special type Hurwitz-zeta values [4] and alternating Euler sums [5]. In this paper we consider the following restricted sum of multiple zeta values…”

Some Weighted Sum Formulas for Multiple Zeta, Hurwitz Zeta, and Alternating Multiple Zeta Values

“…which holds true for arbitrary complex number x (see [26], p. 75 or [28], pp. [12][13][14][15][16][17][18]. e binomial series asserts that for complex number α (see [27], p. 37),…”

“…Furthermore, Zhao [16] used the ideas developed in [15] to evaluate the sums of all multiple Hurwitz zeta values of the depth k and the weight 2n in terms of the Euler numbers. Moreover, Zhao [17] used the theory of symmetric functions to consider the more complicated alternating multiple zeta values and depicted that the sums of all alternating multiple zeta values of the depth k and the weight 2n can be evaluated in terms of the Riemann zeta function and the Euler numbers. More recently, Chen et al [18] used the method of the generating functions to express E(mn, k) by constructing a combinatorial identity of products of the multiple zeta values and the so-called multiple zeta-star values at the repetitions of m, and then used Muneta's [19] and Nakamura's [20] results to reobtain Hoffman's sum formula (10) and confirm Genčev's ( [21], Conjecture 4.1) conjecture on the evaluation of E(4n, k).…”

“…In Section 3, we present some similar weighted sum formulas for the multiple Hurwitz zeta values and deduce Shen and Jia's [22] sum formulas as special cases. Section 4 concentrates on the features that have contributed to the weighted sum formulas for the alternating multiple zeta values, and it then turns out that Zhao's [17] sum formula is obtained in a rather simple way.…”

Unramified Euler sums and Hoffman $\star$ basis

“…The first line is zero (since e0 +e1 +e−1 +e∞ = 0) whereas each other line will contribute by two terms, in order to give (44). Indeed, the projection π L (x), when seeing x as a polynomial (with only even powers) in L m , only keep the constant term; hence, for each term, only one of the exponentials above e x contributes by its linear term i.e.…”