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A Weighted Sum Formula for Alternating Multiple Zeta-Star Values

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Abstract

In the last decade, many authors essentially contributed to the attractive theory of multiple zeta values. Nevertheless, since their introduction in 1992, there are still many hypotheses and open problems waiting to be solved. The aim of this paper is to develop a method for transforming the multiple zeta-star values \(\zeta ^\star (\{2\}_K,c)\) leading to a new sum formula for alternating multiple zeta-star values. Its most simple case has the intelligible form

$$\begin{aligned} \sum _{t=0}^{c-2}(-2)^{t+1} \sum _{\begin{array}{c} i\ge 2,\,\varvec{s}\in \mathbb {N}^t\\ i+|\varvec{s}\!|=c \end{array}} \zeta ^\star ({\overline{i}},\varvec{s}) =(-1)^c\cdot \zeta (c). \end{aligned}$$

As a by-product, we also establish a closed form for a new harmonic-like finite summation containing binomial coefficients.

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Notes

  1. Compare the number of published works in sections C and E on the web page https://www.usna.edu/Users/math/meh/biblio.html collecting publications in the field of multiple zeta values.

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Acknowledgements

The author wishes to thank Minking Eie, Wei-Ming Wang and Orlando Arencibia for valuable discussion and/or support during the course of the work. I am also grateful to Penelope Hamilton for her language suggestions.

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This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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  1. Faculty of Economics, VŠB-Technical University, Sokolská 33, 702 00, Ostrava, Czech Republic

    Marian Genčev

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Genčev, M. A Weighted Sum Formula for Alternating Multiple Zeta-Star Values. Mediterr. J. Math. 18, 236 (2021). https://doi.org/10.1007/s00009-021-01844-z

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