More identities for Fibonacci and Lucas quaternions

Year 2020, Volume: 69 Issue: 1, 369 - 375, 30.06.2020

Nurettin Irmak

https://izlik.org/JA73SD88MP

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}.

Keywords

Fibonacci Quaternions , Lucas Quaternions , recurrence relations

References

• Koshy, T. Fibonacci and Lucas numbers with applications, Wiley, Newyork, 2001.
• Horadam, A. F. Complex Fibonacci numbers and Fibonacci Quaternions, Amer. Math. Monthly, 70 (1963), 289-291.
• Iyer, M. R. A note on Fibonacci Quaternions, The Fib. Quarterly, 3 (1969), 225-229.
• Swamy, M. N. S. On generalized Fibonacci quaternions, The Fib. Quarterly, 5 (1973), 547-550.
• Halıcı, S. On Fibonacci Quaterions, Adv. Appl. Clifford Algebras, 12 (2012), 321-327.
• Akyiğit, M., Kösal, H. H., Tosun, M. Split Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 23 (2013), 535-545.
• Akyiğit, M., Kösal, H. H., Tosun, M. Fibonacci Generalized Quaternions, Adv. Appl. Clifford Algebras, 24 (2014), 631-641.
• Tan, E., Yılmaz, S., Şahin, M. A new generalization of Fibonacci quaternions, Chaos Solitions Fractals, 82 2016, 1-4.
• Tan, E., Yılmaz, S., Şahin, M. A Note on bi-periodic Fibonacci and Lucas quaternions, Chaos Solitions Fractals, 85 2016, 138-142.
• Tan, E., Sahin, M., Yilmaz, S. The generalized bi-periodic Fibonacci quaternions and octonions. Novi Sad J. Math. doi.org/10.30755/NSJOM.07284.
• Morais, J. P., Georgiev, S., Spröbig, W. Real Quaternionic Calculus Handbook, Birkhauser, London, 2014.