Restricted sum formula of alternating Euler sums

Abstract

In this paper, we study restricted sum formulas involving alternating Euler sums which are defined by $$ {\zeta}(s_1,\dots,s_d;{\varepsilon}_1, \dots,{\varepsilon}_d)=\sum_{n_1>\cdots>n_d\ge1} \frac{{\varepsilon}_1^{n_1}\cdots{\varepsilon}_d^{n_d}}{ n_1^{s_1}\cdots n_d^{s_d}}, $$ for all positive integers s1,…,sd and e1=±1,…,ed=±1 with (s1,e1)≠(1,1). We call w=s1+⋯+sd the weight and d the depth. When ej=−1 we say the jth component is alternating. We first consider Euler sums of the following special type: $$ {\xi}(2s_1,\dots,2s_d)={\zeta}\bigl(2s_1, \dots,2s_d;(-1)^{s_1},\dots ,(-1)^{s_d}\bigr). $$ For d≤n, let Ξ(2n,d) be the sum of all ξ(2s1,…,2sd) 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.