On the sum of powers of two $k$-Fibonacci numbers which belongs to the sequence of $k$-Lucas numbers

Year, pages: 2016, 41 - 46

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 con di tions $F_k,0=0$ 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 $F_k,n+1^s+F_k,n-1^s\in (L_k,m)_{m≥ 1}$ for infinitely many positive integers $n$, then $s=1$.

Trojovský, P. 2016. On the sum of powers of two $k$-Fibonacci numbers which belongs to the sequence of $k$-Lucas numbers. In Tatra Mountains Mathematical Publications, vol. 67, no.3, pp. 41-46. 1210-3195.

Trojovský, P. (2016). On the sum of powers of two $k$-Fibonacci numbers which belongs to the sequence of $k$-Lucas numbers. Tatra Mountains Mathematical Publications, 67(3), 41-46. 1210-3195.