Finite and symmetrized colored multiple zeta values - Singer, Johannes; Zhao, Jianqiang (Finite Fields and Their Applications, 2020)
Colored multiple zeta values are special values of multiple polylogarithms evaluated at Nth roots of unity. In this paper, we define both the symmetrized version of these values for all positive integers N and the finite version for N=3,4 and 6. We then show that both versions satisfy the double shuffle relations. Further, for N=3 and 4 we provide strong evidence for an isomorphism connecting the two spaces generated by these two kinds of values. This is a higher level analog of a recent work of Kaneko and Zagier on finite and symmetrized multiple zeta values at level one and of the second author on finite and symmetrized Euler sums at level two.
CRIS-Team Scopus Importer:2020-04-14
Keywords: Colored multiple zeta values; Drinfeld associators; Hopf algebras; Regularization of multiple zeta values

q-Analogues of Multiple Zeta Values and Their Application in Renormalization - Singer, Johannes (Springer Proceedings in Mathematics and Statistics, 2020)
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. 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.
CRIS-Team Scopus Importer:2020-09-11
Keywords: q-Analogues of multiple zeta values; Renormalization; Rota–Baxter operators