Explicit formulas of alternating multiple zeta star values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $

Junjie Quan; Junjie Quan

doi:10.3934/math.2022019

AIMS Mathematics

2022, Volume 7, Issue 1: 288-293. doi: 10.3934/math.2022019

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Explicit formulas of alternating multiple zeta star values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $

Junjie Quan ^,

School of Information Science and Technology, Xiamen University Tan Kah Kee College, Xiamen, Fujian 363105, China

Received: 13 July 2021 Accepted: 06 October 2021 Published: 12 October 2021
MSC : 11A07, 11M32

In a recent paper ^[4], Xu studied some alternating multiple zeta values. In particular, he gave two recurrence formulas of alternating multiple zeta values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $. In this paper, we will give the closed forms representations of $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $ in terms of single zeta values and polylogarithms.
- multiple harmonic (star) sums,
- multiple zeta (star) values,
- multiple polylogarithm function,
- alternating multiple zeta values,
- closed forms
Citation: Junjie Quan. Explicit formulas of alternating multiple zeta star values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $[J]. AIMS Mathematics, 2022, 7(1): 288-293. doi: 10.3934/math.2022019

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Abstract

In a recent paper ^[4], Xu studied some alternating multiple zeta values. In particular, he gave two recurrence formulas of alternating multiple zeta values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $. In this paper, we will give the closed forms representations of $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $ in terms of single zeta values and polylogarithms.

References

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