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Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota–Baxter Algebras

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Periods in Quantum Field Theory and Arithmetic (ICMAT-MZV 2014)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 314))

Abstract

The multiple zeta values (MZVs) have been studied extensively in recent years. Currently there exist a few different types of q-analogs of the MZVs (q-MZVs) defined and studied by mathematicians and physicists. In this paper, we give a uniform treatment of these q-MZVs by considering their double shuffle relations (DBSFs) and duality relations. The main idea is a modification and generalization of the one used by Castillo Medina et al. to a few other types of q-MZVs including the one defined by the author in 2003. With different approach, Takeyama already studied this type by “regularization” and observed that there exist a new family of \({\mathbb Q}\)-linear relations which are not consequences of the DBSFs. We call these duality relations in this paper and generalize them to all other types of q-MZVs. Since there are still some missing relations we further define the most general type of q-MZVs together with a new kind of relations called \(\mathbf{P}\)-\(\mathbf{R}\) relations which are used to lower the deficiencies further. As an application, we will confirm a conjecture of Okounkov on the dimensions of certain q-MZV spaces, either theoretically or numerically, for the weight up to 12. Some relevant numerical data are provided at the end.

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Acknowledgements

This work, supported by NSF grant DMS-1162116, was done while the author was visiting Max Planck Institute for Mathematics and ICMAT at Madrid, Spain. He is very grateful to both institutions for their hospitality and support. He also would like to thank Kurusch Ebrahimi-Fard for a few enlightening conversations and his detailed explanation of their joint paper [7]. The anonymous referees provided many valuable comments and suggestions which greatly improved the clarity of the paper.

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Authors and Affiliations

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

    Jianqiang Zhao

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  1. Jianqiang Zhao
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Correspondence to Jianqiang Zhao .

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Editors and Affiliations

  1. Institute of Mathematical Sciences (ICMAT), Spanish National Research Council (CSIC), Madrid, Spain

    José Ignacio Burgos Gil

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  3. Department of Mathematical Sciences, Durham University, Durham, UK

    Herbert Gangl

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Zhao, J. (2020). Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota–Baxter Algebras. In: Burgos Gil, J., Ebrahimi-Fard, K., Gangl, H. (eds) Periods in Quantum Field Theory and Arithmetic. ICMAT-MZV 2014. Springer Proceedings in Mathematics & Statistics, vol 314. Springer, Cham. https://doi.org/10.1007/978-3-030-37031-2_10

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