Interpolant of truncated multiple zeta functions

We introduce an analytic function $$\Psi (s_1,\ldots ,s_d;w)$$ that interpolates truncated multiple zeta functions $$\zeta _N(s_1,\ldots ,s_d)$$ . We represent this interpolant as a Mellin transform of a function $$G(q_1,\ldots ,q_d;w)$$ 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.