On the sequence made by the linear combination of k-Fibonacci and k-Lucas sequences

TL;DR

This paper introduces a new sequence generated by a linear combination of k-Fibonacci and k-Lucas sequences, explores its properties, identities, and matrix representations, expanding understanding of these generalized sequences.

Contribution

It defines a new sequence S_{k,n} with specific initial conditions and recurrence, and investigates its identities and matrix forms, which is a novel extension of k-Fibonacci and k-Lucas sequences.

Findings

Derived new identities for the sequence S_{k,n}

Analyzed circulant and skew circulant matrices based on S_{k,n}

Extended properties of k-Fibonacci and k-Lucas sequences

Abstract

The Fibonacci sequence is a sequence of numbers that has been studied for hundreds of years. In this paper, we introduce the new sequence S_{k,n} with initial conditions S_{k,0} = 2b and S_{k,1} = bk + a, which is generated by the recurrence relation S_{k,n} = kS_{k,n-1} +S{k,n-2} for n >= 2, where a, b, k are real numbers. Using the sequence S_{k,n}, we introduce and prove some special identities. Also, we deal with the circulant and skew circulant matrices for the sequence S_{k,n}.