The sequences of Fibonacci and Lucas for each real quadratic fields $\mathbb{Q}(\sqrt{d})$

Abstract

We construct the sequences of Fibonacci and Lucas at any quadratic field $\mathbb{Q}(\sqrt{d})$ with $d>0$ square free, noting in general that the 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 $\mathbb{Q}(\sqrt{d})$ where, under certain conditions, some of this constructions correspond to $k$-Fibonacci sequence for some $k\in \mathbb{N}$. Of course, for both sequences, we obtain the generating function, Golden ratio, Binet's formula and some identities that they keep.

17 pages, 2 table