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

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.

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

“…By adding the two results above, we obtain Q(x) as 3 1 − (4ab + 2)x 2 + x 4 which completes the proof. Theorem 2.…”

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.…”