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On the convolutions of sums of multiple zeta(-star) values of height one

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Abstract

In this paper, we investigate the sums of multiple zeta(-star) values of height one: \(Z_{\pm }(n)=\sum _{a+b=n} (\pm 1)^b\zeta (\{1\}^a,b+2)\), \(Z_{\pm }^{\star }(n)=\sum _{a+b=n} (\pm 1)^b\zeta ^{\star }(\{1\}^a,b+2)\). In particular, we prove that the weighted sum

$$\begin{aligned}\sum _{\begin{array}{c} 0\le m\le p\\ m: \mathrm{even} \end{array}} \sum _{\mid \varvec{\alpha }\mid =p+3} 2^{\alpha _{m+1}+1}\zeta (\alpha _0,\alpha _1,\ldots ,\alpha _m,\alpha _{m+1}+1) \end{aligned}$$

can be evaluated through the convolution of \(Z_{-}(m)\) and \(Z_{+}(n)\) with \(m+n=p\).

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Authors and Affiliations

  1. Department of Mathematics, University of Taipei, No. 1, Ai-Guo West Road, Taipei, 100234, Taiwan

    Kwang Wu Chen & Minking Eie

  2. Department of Mathematics, National Chung Cheng University, 168 University Road, Min-Hsiung, Chia-Yi, 62145, Taiwan

    Kwang Wu Chen & Minking Eie

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  1. Kwang Wu Chen
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The first author was funded by the Ministry of Science and Technology, Taiwan, R. O. C., under Grant MOST 110-2115-M-845-001.

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Chen, K.W., Eie, M. On the convolutions of sums of multiple zeta(-star) values of height one. Ramanujan J 59, 1197–1223 (2022). https://doi.org/10.1007/s11139-022-00628-7

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