#### 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 *j*th 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*(2*n*,*d*) be the sum of all *ξ*(2*s* 1,…,2*s* *d* ) of fixed weight 2*n* and depth *d*. We derive a formula for *Ξ*(2*n*,*d*) using the theory of symmetric functions established by Hoffman recently. We also consider restricted sum formulas of Euler sums with fixed weight 2*n*, 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.

#### Recommended Citation

Zhao, Jianqiang.
2015.
"Restricted Sum Formula of Alternating Euler Sums."
*The Ramanujan Journal*, 36 (3): 375-401.
doi: 10.1007/s11139-013-9533-8

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/414

## Comments

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The Ramanujan Journal.