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

Let k ≥ 1 and denote (F_k,n)_n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation F_k,n = kF_k,n−1 +F_k,n−2, with initial conditions F_k,0 = 0 and F_k,1 = 1. In the same way, the k-Lucas sequence (L_k,n)_n≥0 is defined by satisfying the same recurrence relation with initial values L_k,0 = 2 and L_k,1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that F_k,n+1 + F_k,n−1 = L_k,n, for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and Fk,n+1s+Fk,n−1s∈(Lk,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.