Renormalization of Multiple $q$-Zeta Values

Details Renormalization of Multiple $q$-Zeta Values Renormalization of Multiple $q$-Zeta Values

2006-12-04

arxiv.org/abs/math/0612093v3

Acta Mathematica Sinica, English Series, 24 (2008), pp. 1593-1616.

Mathematics

Number Theory

22 pages. This is a substantial revision of the first version. I provide a new and complete proof of the fact that our renorma

Scientific paper

10.1007/s10114-008-6646-x

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 $\zeta_q(s_1,...,s_d)$ 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 $\zeta_q(s_1,...,s_d)$ (i.e., $s_1\le 1$). Moreover, when $\qup$ our renormalizations agree with those of Guo and Zhang.

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