The Lerch-Type Zeta Function of a Recurrence Sequence of Arbitrary Degree

Abstract

[EN]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.