  We obtain several expansions for $\zeta(s)$ involving a sequence of
polynomials in $s$, denoted in this paper by $\alpha_k(s)$. These polynomials
can be regarded as a generalization of Stirling numbers of the first kind and
our identities extend some series expansions for the zeta function that are
known for integer values of $s$. The expansions also give a different approach
to the analytic continuation of the Riemann zeta function.
