The Sequences of Fibonacci and Lucas for Real Quadratic Number Fields

We construct the sequences of Fibonacci and Lucas in any quadratic field Q(root 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 Q(root d) and then we observe the generalizations for any unit of Q(root d). Under certain conditions some of these constructions correspond to k-Fibonacci sequence for some k is an element of N. Further, for both sequences, we obtain the generating function, Golden ratio, Binet's formula and some identities that they keep.