Interpolant of truncated multiple zeta functions

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