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 \(\varphi (z,s,x)\) on x and z, and its properties as a function of three complex variables.
