Restricted Sum Formula of Alternating Euler Sums

TL;DR

This paper derives formulas for sums of alternating Euler sums with fixed weight and depth, utilizing symmetric function theory, and explores cases with a fixed number of alternating components.

Contribution

It introduces new restricted sum formulas for alternating Euler sums, especially for specific numbers of alternating components, expanding understanding of their structure.

Findings

Derived a formula for (2n,d) using symmetric functions.

Determined exact formulas for cases with =1 or =d.

Analyzed cases with <5 for more complex scenarios.

Abstract

In this paper we study restricted sum formulas involving alternating Euler sums which are defined by \zeta(s_1,...,s_{d};\epsilon_1,...,\epsilon_d)=\sum_{n_1>...>n_d\ge 1}\frac{\epsilon_1^{n_1}... \epsilon_{d}^{n_d}}{n_1^{s_1}... n_d^{s_d}}, for all positive integers s_1,...,s_{d} and \epsilon_1=\pm 1,..., \epsilon_{d}=\pm 1 with (s_1,\epsilon_1) unequal (1,1). We call w=s_1+...+s_{d} the weight and d the depth. When \epsilon_j=-1 we say the jth component is alternating. We first consider Euler sums of the following special type: \xi(2s_1,...,2s_{d})=\zeta(2s_1,...,2s_{d};(-1)^{s_1},...,(-1)^{s_{d}}). For d\le n, let \Xi(2n,d) be the sum of all \xi(2s_1,..., 2s_{d}) of fixed weight 2n and depth d. We derive a formula for \Xi(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 \alpha of alternating components at even arguments. When \alpha=1 or \alpha=d we can determine precisely the restricted sum formulas. For other \alpha we only treat the cases d<5 completely since the symmetric function theory becomes more and more unwieldy to work with when \alpha moves closer to d/2.