2019/04/30 by Lam-Estrada, Pablo, Maldonado-Ramírez, Myriam Rosalía, López-Bonilla, José Luis +1
#FOS: Mathematics #Number Theory (math.NT) #Primary 11B39 #Secundary 11R11 and 11R04
paper · doi:10.48550/arxiv.1904.13002
We construct the sequences of Fibonacci and Lucas at any quadratic field ℚ(√(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 ℚ(√(d) ) and then we observe the generalizations for any unit of ℚ(√(d) ) where, under certain conditions, some of this constructions correspond to k-Fibonacci sequence for some k∈ ℕ. Of course, for both sequences, we obtain the generating function, Golden ratio, Binet's formula and some identities that they keep.