A.B. Goncharov

B. Berndt

E. Grosswald

G. Shimura

H. Gangl

I. Macdonald

J. Zhao

Jianqiang Zhao

M. Petkovsek

P. Deligne

T. Nakamura

Z. Shen

English

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The Ramanujan Journal

Restricted sum formula of alternating Euler sums

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2 Restricted Sum Formula of Alternating Euler Sums

In this paper, we study restricted sum formulas involving alternating Euler sums which are defined by    ζ(s1,…,sd;ε1,…,εd)=∑n1\u3e⋯\u3end≥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 ) 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\u3c5 completely since the symmetric function theory becomes more and more unwieldy to work with when α moves closer to d/2

Zhao, Jianqiang

Georgia Southern University: Digital Commons@Georgia Southern

Restricted Sum Formula of Alternating Euler Sums

Zhao, J.

MPG.PuRe

https://digitalcommons.georgiasouthern.edu/cgi/viewcontent.cgi?article=1331&context=math-sci-facpubs

Restricted sum formula of alternating Euler sums

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Georgia Southern University: Digital Commons@Georgia Southern

MPG.PuRe