q-Analogues of Multiple Zeta Values and Their Application in Renormalization

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


In this paper we report on recent developments on q-analogues of multiple zeta values (MZVs), which are power series in a formal parameter q that reduce to classical MZVs in the limit \(q\rightarrow 1\). First of all, we systematically develop the double shuffle relations of three q-models, whose shuffle products rely on a description of iterated Rota–Baxter operators. In the second part we use two of these q-models to construct solutions to the renormalization problem of MZVs, i.e., a systematic extension of MZVs to negative integers. In one case the renormalized MZVs satisfy the quasi-shuffle relations and in the other case the shuffle relations are verified.


q-Analogues of multiple zeta values Rota–Baxter operators Renormalization 



Some results presented in this work are based on a project which was carried out during the Research Trimester on Multiple Zeta Values, Multiple Polylogarithms, and Quantum Field Theory (September–December 2014) at ICMAT, Madrid. I have greatly enjoyed the hospitality and the nice and stimulating atmosphere at ICMAT. Furthermore I am very grateful to Andreas Knauf for his comments which significantly improved the paper.


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Authors and Affiliations

  1. 1.Department MathematikFriedrich–Alexander–Universität Erlangen–NürnbergErlangenGermany

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