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arXiv · 2303.16602

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

Álvaro Serrano Holgado·Luis Manuel Navas Vicente
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Abstract

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 $z$.

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Álvaro Serrano Holgado, Luis Manuel Navas Vicente. 2023-03-29. The Lerch-type zeta function of a recurrence sequence of arbitrary degree. https://arxiv.org/abs/2303.16602

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