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 wellknown matrix sequences, such as Fibonacci, Pell, k-Fibonacci matrix sequences are special cases of this generalized matrix sequence.

On the bi-periodic Padovan sequences

“…Irmak et al have presented various studies on periodic functions [1,14,15]. Various identities have been generalized by many researchers [2,4,6,7,19,[21][22][23][24][25]27].…”

A New Generalization of Jacobsthal Lucas Numbers (Bi-Periodic Jacobsthal Lucas Sequence)

“…He also found some interesting identities between the above two sequences. The authors in [8], [9], [10], [11], [12], [13], [14], [15] gave interesting properties of bi-periodic sequences.…”

“…α and β are the roots of the nonlinear quadratic equation for the bi-periodic Jacobsthal sequence which is given as x 2 − abx − 2ab = 0. In [8], [9], [11] the authors carried bi-periodic sequences to bi-periodic Fibonacci, Lucas and Jacobsthal matrix sequences. The authors, in [12] gave interesting properties of bi-periodic Jacobsthal and bi-periodic Jacobsthal-Lucas sequences.…”

Matrix Representation of Bi-Periodic Jacobsthal Sequence

“…The direct relationship between the bi-periodic Jacobsthal and the bi-periodic Jacobsthal Lucas sequences were obtained as C n = 2 n−1 + n+1 and (ab + 8) n = 2C n−1 + C n+1 . In [16], Coskun and Taskara defined the bi-periodic Fibonacci matrix sequence as…”