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Renormalization of Multiple $q$-Zeta Values

Jianqiang Zhao
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Abstract

In this paper we shall define the renormalization of the multiple $q$-zeta values (M$q$ZV) which are special values of multiple $q$-zeta functions $ζ_q(s_1,...,s_d)$ when the arguments are all positive integers or all non-positive integers. This generalizes the work of Guo and Zhang (math.NT/0606076v3) on the renormalization of Euler-Zagier multiple zeta values. We show that our renormalization process produces the same values if the M$q$ZVs are well-defined originally and that these renormalizations of M$q$ZV satisfy the $q$-stuffle relations if we use shifted-renormalizations for all divergent $ζ_q(s_1,...,s_d)$ (i.e., $s_1\le 1$). Moreover, when $\qup$ our renormalizations agree with those of Guo and Zhang.

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Jianqiang Zhao. 2007-05-26. Renormalization of Multiple $q$-Zeta Values. https://doi.org/10.1007/s10114-008-6646-x

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