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arXiv · 2209.03023

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

Lejla Smajlović·Zenan Šabanac·Lamija Šćeta
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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 $\Re(s)\leq 0$.

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Lejla Smajlović, Zenan Šabanac, Lamija Šćeta. 2022-09-07. On the Hurwitz-type zeta function associated to the Lucas sequence. https://arxiv.org/abs/2209.03023

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