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Variants of multiple zeta values with even and odd summation indices

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Abstract

In this paper, we define and study a variant of multiple zeta values (MZVs) of level two, called multiple mixed values or multiple M-values (MMVs), which forms a subspace of the space of alternating MZVs. This variant includes both Hoffman’s multiple t-values and Kaneko–Tsumura’s multiple T-values as special cases. We set up the algebraic framework for the double shuffle relations of the MMVs, and exhibits nice properties similar to ordinary MZVs such as the duality, integral shuffle and series stuffle relations. We also investigate several T-variants of Kaneko–Yamamoto type MZVs by establishing some explicit relations between these T-variants and Kaneko–Tsumura \(\psi \)-values, and prove that all the \(\psi \)-values can be expressed in terms of multiple T-values. In the end, we discuss the explicit evaluations of a kind of MMVs at depth two and three and compute the dimensions of a few interesting subspaces of MMVs for weight less than 13.

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Acknowledgements

CX expresses his deep gratitude to Prof. Masanobu Kaneko and Prof. Weiping Wang for valuable discussions and comments. JZ wants to thank Prof. Kaneko for inviting him to visit the Multiple Zeta Research Center at Kyushu University where this joint work was started. Both authors thank the anonymous referee for many invaluable comments and suggestions which have improved the paper greatly. This work was supported by the Scientific Research Foundation for Scholars of Anhui Normal University, the National Natural Science Foundation of China (Grant No. 12101008), the Natural Science Foundation of Anhui Province (Grant No. 2108085QA01) and the University Natural Science Research Project of Anhui Province (Grant No. KJ2020A0057).

Funding

Scientific Research Foundation for Scholars of Anhui Normal University, the National Natural Science Foundation of China (Grant no. 12101008), the Natural Science Foundation of Anhui Province (Grant no. 2108085QA01) and the University Natural Science Research Project of Anhui Province (Grant no. KJ2020A0057).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Anhui Normal University, Wuhu, 241002, People’s Republic of China

    Ce Xu

  2. Department of Mathematics, The Bishop’s School, La Jolla, CA, 92037, USA

    Jianqiang Zhao

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  1. Ce Xu
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  2. Jianqiang Zhao
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Correspondence to Jianqiang Zhao.

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Dedicated to Professor Masanobu Kaneko on the occasion of his 60th birthday.

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Xu, C., Zhao, J. Variants of multiple zeta values with even and odd summation indices. Math. Z. 300, 3109–3142 (2022). https://doi.org/10.1007/s00209-021-02889-2

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