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

  • Jianqiang ZhaoEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 314)


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.


Multiple zeta values q-analog of multiple zeta values Double shuffle relations Duality relations Rota-Baxter algebras 



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. 1.Department of MathematicsThe Bishop’s SchoolLa JollaUSA

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