Renormalization of Multiple $q$-Zeta Values

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 ${\zeta }_q(s_1,...,s_d)$ 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.

22 pages. This is a substantial revision of the first version. I provide a new and complete proof of the fact that our renormalizations satisfy the q-stuffle relations using the shifting principle of MqZVs

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Cited by in corpus (6)

• Rota-Baxter Algebras and Dendriform Algebras
• q-Multiple Zeta Functions and q-Multiple Polylogarithms
• Algebraic Birkhoff decomposition and its applications
• The Hopf algebra of (q)multiple polylogarithms with non-positive arguments
• Differential Birkhoff decomposition and the renormalization of multiple zeta values
• Renormalisation of q-regularised multiple zeta values