The Lerch-type zeta function of a recurrence sequence of arbitrary degree

We consider the series ${\sum }_{n=1}^{\mathrm{\infty }}{z}^n(a_n+x{)}^{-s}$ 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 $\phi (z,s,x)$ 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 $\phi$, which turn out to be polynomials in $x$. In addition we study the dependence of $\phi$ on $x$ and $z$.