The Matrix Sequence in terms of bi-periodic Fibonacci numbers

TL;DR

This paper introduces a bi-periodic Fibonacci matrix sequence that generalizes known matrix sequences, explores its properties, and connects it to classical sequences like Fibonacci and Pell, providing new insights into their behaviors.

Contribution

It defines a new bi-periodic Fibonacci matrix sequence and investigates its generating function, Binet formula, and summations, unifying several classical matrix sequences as special cases.

Findings

Derived the generating function and Binet formula for the sequence.

Established connections between the bi-periodic Fibonacci matrix sequence and classical sequences.

Showed that known matrix sequences are special cases of the new generalized sequence.

Abstract

In this paper, we define the bi-periodic Fibonacci matrix sequence that represent bi-periodic Fibonacci numbers. Then, we investigate generating function, Binet formula and summations of bi-periodic Fibonacci matrix sequence. After that, we say that some behaviours of bi-periodic Fibonacci numbers also can be obtained by considering properties of this new matrix sequence. Finally, we express that well-known matrix sequences, such as Fibonacci, Pell, $k$-Fibonacci matrix sequences are special cases of this generalized matrix sequence.