On $k$-Fibonacci sums by matrix methods

TL;DR

This paper derives sums of $k$-Fibonacci and $k$-Lucas numbers using matrix methods, providing novel proofs for known identities that are not previously documented in the literature.

Contribution

It introduces a new matrix-based proof technique for identities involving $k$-Fibonacci and $k$-Lucas numbers, which is not present in existing literature.

Findings

Derived sums of $k$-Fibonacci and $k$-Lucas numbers using matrices

Provided new proofs for known identities with matrix methods

Enhanced understanding of matrix approaches in Fibonacci number identities

Abstract

In this paper, some $k$-Fibonacci and $k$-Lucas with arithmetic indexes sums are derived by using the matrices $R_{a}=\left[ \begin{array}{lr} L_{k,a} & -(-1)^{a} \\ 1 & 0 \end{array}\right]$ and $S_{a}=\frac{1}{2}\left[ \begin{array}{lr} L_{k,a} & \Delta_{a} \\ 1 & L_{k,a} \end{array}\right]$, where $\Delta_{a}=L_{k,a}^{2}-4(-1)^{a}$. 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.