Abstract
In this paper, we define the associate matrix as% \begin{equation*} F=\left( \begin{array}{cc} 1+i+2j+3k & i+j+2k \\ i+j+2k & 1+j+k% \end{array}% \right) . \end{equation*}% By the means of the matrix $F,$ we give several identities about Fibonacci and Lucas quaternions by matrix methods. Since there are two different determinant definitions of a quaternion square matrix (whose entries are quaternions), we obtain different Cassini identities for Fibonacci and Lucas quaternions apart from Cassini identities that given in the papers \cite% {halici} and \cite{akyigit2}.
Highlights
The quaternions were described by Irish mathematicians Sir William and Rowan Hamilton as a extension of a complex number
We present some novel identities between Fibonacci and Lucas quaternions by using matrix method
First theorem is about the Cassini identity belongs to Fibonacci and Lucas quaternions
Summary
The quaternions were described by Irish mathematicians Sir William and Rowan Hamilton as a extension of a complex number. There are lots of amazing identities belongs to Fibonacci and Lucas numbers. Horadam [2] de...ned nth Fibonacci and Lucas quaternions as follows, Qn = Fn + iFn+1 + jFn+2 + kFn+3 and Kn = Ln + iLn+1 + jLn+2 + kLn+3: The conjugates of these quaternions are given by Afterwards, they gave Fibonacci generalized quaternions and they used the well-known identities related to the Fibonacci and Lucas numbers to obtain the relations regarding these quaternions in [7]. They de...ned bi-periodic Fibonacci and Lucas quaternions.
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