  We give a new and very concise proof of the existence of a holomorphic
continuation for a large class of twisted multivariable zeta functions. To do
this, we use a simple method of "decalage" that avoids using an integral
representation of the zeta function. This allows us to derive explicit
recurrence {\it relations} between the values at $T-$tuples of negative
integers. This also extends some earlier results of several authors where the
underlying polynomials were products of linear forms.
