  We introduce an analytic function $\Psi(s_1,\ldots,s_r;w)$ that interpolates
truncated multiple zeta functions $\zeta_N(s_1,\ldots,s_r)$. We represent this
interpolant as a Mellin transform of a function $G(q_1,\ldots,q_r;w)$ and,
using this expression, give the analytic continuation. Further, the harmonic
product relations for $\Psi$ and $G$ are established via relevant Hopf algebra
structures, and some properties of the function $G$ are provided.
