Multiple $q$-Zeta Values

We introduce a $q$-analog of the multiple harmonic series commonly referred to as multiple zeta values. The multiple $q$-zeta values satisfy a $q$-stuffle multiplication rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values. Additionally, multiple $q$-zeta values can be viewed as special values of the multiple $q$-polylogarithm, which admits a multiple Jackson $q$-integral representation whose limiting case is the Drinfel'd simplex integral for the ordinary multiple polylogarithm when $q=1$. The multiple Jackson $q$-integral representation for multiple $q$-zeta values leads to a second multiplication rule satisfied by them, referred to as a $q$-shuffle. Despite this, it appears that many numerical relations satisfied by ordinary multiple zeta values have no interesting $q$-extension. For example, a suitable $q$-analog of Broadhurst's formula for $\zeta(\{3,1\}^n)$, if one exists, is likely to be rather complicated. Nevertheless, we show that a number of infinite classes of relations, including Hoffman's partition identities, Ohno's cyclic sum identities, Granville's sum formula, Euler's convolution formula, Ohno's generalized duality relation, and the derivation relations of Ihara and Kaneko extend to multiple $q$-zeta values.