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arXiv · 1904.13002

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

Pablo Lam-Estrada·Myriam Rosalía Maldonado-Ramírez·José Luis López-Bonilla·Fausto Jarquín-Zárate
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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.

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Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, Fausto Jarquín-Zárate. 2019-04-30. The sequences of Fibonacci and Lucas for each real quadratic fields $\mathbb{Q}(\sqrt{d}\ )$. https://arxiv.org/abs/1904.13002

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