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Restricted Sum Formula Of Alternating Euler Sums, Jianqiang Zhao
Restricted Sum Formula Of Alternating Euler Sums, Jianqiang Zhao
Department of Mathematical Sciences Faculty Publications
In this paper, we study restricted sum formulas involving alternating Euler sums which are defined by ζ(s1,…,sd;ε1,…,εd)=∑n1>⋯>nd≥1εn11⋯εnddns11⋯nsdd,
for all positive integers s 1,…,s d and ε 1=±1,…,ε d =±1 with (s 1,ε 1)≠(1,1). We 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: ξ(2s1,…,2sd)=ζ(2s1,…,2sd;(−1)s1,…,(−1)sd).
For d≤n, let Ξ(2n,d) be the sum of all ξ(2s 1,…,2s d ) …