Matrix Representation of Bi-Periodic Jacobsthal Sequence

Abstract

In this paper, we bring into light the matrix representation of biperiodic Jacobsthal sequence, which we shall call the bi-periodic Jacobsthal Matrix sequence. We define it asWe obtained the nth general term of this new matrix sequence. By studying the properties of this new matrix sequence, the well-known Cassini or Simpson's formula was obtained. We then proceeded to find its generating function as well as the Binet formula. Some new properties and two summation formulas for this new generalized matrix sequence are also given.By using Lemma 1, we obtained that;Similarly, the odd part of the above series is simplified as follows

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.…”

“…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. Uygun and Karatas, in [17] introduced bi-periodic Pell-Lucas sequence.…”

On Bi-periodic Jacobsthal and Jacobsthal-Lucas Quaternions

“…In (Uygun & Owusu, 2017), some relationships between these numbers were obtained as = 2̂− 1 +̂+ 1 and ( + 8)̂= 2 −1 + +1 . The generating function of the sequence *̂+ is ( ) = (1+ −2 2 ) 1−( +4) 2 +4 4 and the Binet formula of the sequence *̂+ is ̂= (…”