On the Hurwitz-type zeta function associated to the Lucas sequence

TL;DR

This paper investigates the properties of the theta and Hurwitz-type zeta functions linked to the Lucas sequence, deriving asymptotic expansions, meromorphic continuation, and pole residues to deepen understanding of these special functions.

Contribution

It provides the first detailed analysis of the Hurwitz-type zeta function associated with Lucas sequences, including asymptotic behavior and pole structure.

Findings

Asymptotic expansion of the theta function as t approaches 0.

Meromorphic continuation of the zeta function to the entire complex plane.

Identification of residues at all poles in the left half-plane.

Abstract

We study the theta function and the Hurwitz-type zeta function associated to the Lucas sequence $U=\{U_n(P,Q)\}_{n\geq 0}$ of the first kind determined by the real numbers $P,Q$ under certain natural assumptions on $P$ and $Q$. We deduce an asymptotic expansion of the theta function $\theta_U(t)$ as $t\downarrow 0$ and use it to obtain a meromorphic continuation of the Hurwitz-type zeta function $\zeta_{U}\left( s,z\right) =\sum\limits_{n=0}^{\infty }\left(z+U_{n}\right) ^{-s}$ to the whole complex $s-$plane. Moreover, we identify the residues of $\zeta_{U}\left( s,z\right)$ at all poles in the half-plane $\Re(s)\leq 0$.