On $k$-Fibonacci sums by matrix methods

Abstract: In this paper, some $k$-Fibonacci and $k$-Lucas with arithmetic indexes sums are derived by using the matrices $$R_{a}=\left[ \begin{array}{lr} L_{k,a} &amp; -(-1)^{a} \ 1 &amp; 0 \end{array}\right]$$ and $$S_{a}=\frac{1}{2}\left[ \begin{array}{lr} L_{k,a} &amp; \Delta_{a} \ 1 &amp; L_{k,a} \end{array}\right]$$, where $\Delta_{a}=L_{k,a}<sup>{2}-4(-1)<sup>{a}$. The most notable side of this paper is our proof method, since all the identities used in the proofs of main theorems are proved previously by using the matrices $R_{a}$ and $S_{a}$, with $a$ an natural number. Although the identities we proved are known, our proofs are not encountered in the $k$-Fibonacci and $k$-Lucas numbers literature.