Document Type : Regular research papers
Authors
1
Matematik Bölümü, Fen-Edebiyat Fakültesi, Mersin Üniversitesi, Çiftlikköy Kampüsü, TR-33343, Mersin, Türkiye
2
Matematik Bölümü, Fen-Edebiyat Fakültesi, Mersin Üniversitesi, Çiftlikköy Kampüsü, TR-33343, Mersin, Türkiye
Abstract
Ferns introduced pseudo-Fibonacci and pseudo-Lucas sequences in 1968 as novel generalizations of the Fibonacci and Lucas sequences as follows:\\
First, consider two recurrence relations
\begin{align}\label{d1}
\Phi_{n+1}=\Phi_{n}+\Psi_{n},
\end{align}
\begin{align}\label{d2}
\Psi_{n+1}=\Phi_{n+1}+\gamma\Phi_{n}
\end{align}
with initial conditions $\Phi_1=1$ and $\Psi_1=1$ in which $\gamma$ is a positive integer. $\Phi$ and $\Psi$ are pseudo-Fibonacci and pseudo-Lucas numbers, respectively (see \cite{Ferns}). Actually, by eliminating first the $\Phi$s and then the $\Psi$s, from (\ref{d1}) and (\ref{d2}), the following pseudo-Fibonacci and pseudo-Lucas sequences are obtained
\begin{align}\label{d3}
\Phi_{n+2}=2\Phi_{n+1}+\gamma\Phi_{n},
\end{align}
\begin{align}\label{d4}
\Psi_{n+2}=2\Psi_{n+1}+\gamma\Psi_{n}
\end{align}
with initial conditions $\Phi_0=0$, $\Phi_1=1$ and $\Psi_0=1$, $\Psi_1=1$, respectively. There are a lot of quaternion numbers that are related to the Fibonacci and Lucas numbers or their generalizations have been described and extensively explored. The coefficients of these quaternions have been chosen from terms of Fibonacci and Lucas numbers. In this study, we define two new quaternions that are pseudo-Fibonacci and pseudo-Lucas quaternions. Then, we give their Binet-like formula, generating functions, certain binomial sums and Honsberg-like, d'Ocagne-like, Catalan-like and Cassini-like identities.
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