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

## TL;DR

This paper generalizes Fibonacci and Lucas sequences to real quadratic fields using fundamental units, deriving their properties, generating functions, and identities, extending classical results to these algebraic structures.

## Contribution

It introduces a construction of Fibonacci and Lucas sequences in quadratic fields using fundamental units, generalizing classical sequences and their properties.

## Key findings

- Sequences constructed for quadratic fields with properties similar to classical Fibonacci and Lucas sequences.
- Derived generating functions, Golden ratio, and Binet's formulas for these sequences.
- Identified conditions under which these sequences relate to k-Fibonacci sequences.

## 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.