A note on the bi-periodic Fibonacci and Lucas matrix sequences

TL;DR

This paper introduces the bi-periodic Lucas matrix sequence, explores its properties, and examines its relationship with bi-periodic Fibonacci matrices, showing how various known sequences are special cases.

Contribution

The paper presents a new generalized bi-periodic Lucas matrix sequence and analyzes its fundamental properties and connections to other matrix sequences.

Findings

Established properties of the bi-periodic Lucas matrix sequence

Derived relationships between bi-periodic Fibonacci and Lucas matrix sequences

Showed that known sequences are special cases of the new sequence

Abstract

In this paper, we introduce the bi-periodic Lucas matrix sequence and present some fundamental properties of this generalized matrix sequence. Moreover, we investigate the important relationships between the bi-periodic Fibonacci and Lucas matrix sequences. We express that some behaviours of bi-periodic Lucas numbers also can be obtained by considering properties of this new matrix sequence. Finally, we say that the matrix sequences as Lucas, $k$-Lucas and Pell-Lucas are special cases of this generalized matrix sequence.