The Matrix Sequence in terms of bi-periodic Fibonacci numbers

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.