The Sequences of Fibonacci and Lucas for Quadratic Fields

Abstract

We construct the sequences of Fibonacci and Lucas in any quadratic
field $\mathbb{Q}(\sqrt{d}\,)$ with $d>0$ square free, noting that
the general properties remain valid as those given by the
classical sequences of Fibonacci and Lucas for the case $d = 5$,
under the respective variants. For this construction, we use the
fundamental unit of $\mathbb{Q}(\sqrt{d}\,)$ and then we observe the
generalizations for any unit of $\mathbb{Q}(\sqrt{d}\,)$. Under
certain conditions some of these constructions correspond to
$k$-Fibonacci sequence for some $k\in \mathbb{N}$. Further, for
both sequences, we obtain the generating function, Golden ratio,
Binet's formula and some identities that they keep.