On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

Abstract

Abstract Let k ≥ 1 and denote (Fk,n ) n≥ 0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n− 1 +Fk,n− 2, with initial conditions Fk, 0 = 0 and Fk, 1 = 1. In the same way, the k-Lucas sequence (Lk,n ) n≥ 0 is defined by satisfying the same recurrence relation with initial values Lk, 0 = 2 and Lk, 1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that Fk,n +1 + Fk,n− 1 = Lk,n , for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and F k , n + 1 s + F k , n − 1 s ∈ ( L k , m ) m ≥ 1 $F_{k,n + 1}^s + F_{k,n - 1}^s \in \left( {L_{k,m} } \right)_{m \ge 1} $ for infinitely many positive integers n, then s =1.