Document Type

Article

Publication Date

4-2015

Publication Title

The Ramanujan Journal

DOI

10.1007/s11139-013-9533-8

Abstract

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 dn, 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.

Comments

This version of the paper was obtained from arXIV.org. In order for the work to be deposited in arXIV.org, the authors must hold the rights or the work must be under Creative Commons Attribution license, Creative Commons Attribution-Noncommercial-ShareAlike license, or Create Commons Public Domain Declaration. The publisher's final edited version of this article is available at The Ramanujan Journal.

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