Let $k\geq 1$ and denote $(F_{k,n})_{n\geq 0}$, 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 $(L_{k,n})_{n\geq 0}$ is defined by satisfying the same recurrence relation with initial values $L_{k,0}=2$ and $L_{k,1}=k$. These sequences was introduced by Falcon and Plaza and they showed many of its properties too. In particular, they proved that $F_{k,n+1}+F_{k,n-1}=L_{k,n}$, for all $k\geq 1$ and $n\geq 0$. In this paper, we shall prove that if $k\geq1$ and $F_{k,n+1}^s+F_{k,n-1}^s\in (L_{k,m})_{m\geq 1}$ for infinitely many positive integers $n$, then $s=1$.
@article{454,
title = {On the sum of powers of two $k$-Fibonacci numbers which belongs to the sequence of $k$-Lucas numbers},
journal = {Tatra Mountains Mathematical Publications},
volume = {65},
year = {2016},
doi = {10.2478/tatra.v67i0.454},
language = {EN},
url = {http://dml.mathdoc.fr/item/454}
}
Trojovský, Pavel. On the sum of powers of two $k$-Fibonacci numbers which belongs to the sequence of $k$-Lucas numbers. Tatra Mountains Mathematical Publications, Tome 65 (2016) . doi : 10.2478/tatra.v67i0.454. http://gdmltest.u-ga.fr/item/454/