  We consider the series $\sum_{n=1}^{\infty} z^{n} (a_{n} + x)^{-s}$ where
$a_{n}$ satisfies a linear recurrence of arbitrary degree with integer
coefficients. Under appropriate conditions, we prove that it can be continued
to a meromorphic function on the complex $s$-plane. Thus we may associate a
Lerch-type zeta function $\varphi(z,s,x)$ to a general recurrence. This
subsumes all previous results which dealt only with the ordinary zeta and
Hurwitz cases and degrees $2$ and $3$. Our method generalizes a formula of
Ramanujan for the classical Hurwitz-Riemann zeta functions. We determine the
poles and residues of $\varphi$, which turn out to be polynomials in $x$. In
addition we study the dependence of $\varphi$ on $x$ and $z$.
