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\vskip 1cm{\LARGE\bf
A Note on Catalan's Identity for the $k$-Fibonacci Quaternions \\
\vskip .1in
}
\vskip 1cm
\large
Emrah Polatl\i\ and Seyhun Kesim\\
Department of Mathematics \\
B\"{u}lent Ecevit University\\
67100 Zonguldak\\
Turkey\\
\href{mailto:emrahpolatli@gmail.com}{\tt emrahpolatli@gmail.com} \\
\href{mailto:seyhun.kesim@beun.edu.tr}{\tt seyhun.kesim@beun.edu.tr} \\
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\begin{abstract}
Ram\'{\i}rez recently conjectured a version of Catalan's identity for
the $k$-Fibonacci quaternions.  In this note we 
give a proof of (a suitably reformulated version of) this identity.
\end{abstract}

\section{Introduction}

For any positive real number $k$, define the $k$-Fibonacci and $k$-Lucas sequences,
$\left( F_{k,n}\right) _{n\in
\mathbb{N}
}$ and $\left( L_{k,n}\right) _{n\in
\mathbb{N}
}$, as follows:
\begin{equation*}
F_{k,0}=0,\text{ }F_{k,1}=1,\text{ and }F_{k,n}=kF_{k,n-1}+F_{k,n-2},\text{ }
n\geq 2,
\end{equation*}
and
\begin{equation*}
L_{k,0}=2,\text{ }L_{k,1}=k,\text{ and }L_{k,n}=kL_{k,n-1}+L_{k,n-2},\text{ }
n\geq 2,
\end{equation*}
respectively.

Let $\alpha $ and $\beta $ be the roots of the characteristic equation $
x^{2}-kx-1=0$. Then the Binet formulas for the $k$-Fibonacci and $k$-Lucas
sequences are
\begin{equation*}
F_{k,n}=\frac{\alpha ^{n}-\beta ^{n}}{\alpha -\beta }
\end{equation*}
and
\begin{equation*}
L_{k,n}=\alpha ^{n}+\beta ^{n},
\end{equation*}
where $\alpha =\left( k+\sqrt{k^{2}+4}\right) /2$ and $\beta =\left( k-\sqrt{
k^{2}+4}\right) /2$.

A {\it quaternion} $p$, with real components $a_{0}$, $a_{1}$, $a_{2}$, $
a_{3}$ and basis $\boldsymbol{1}$, $\boldsymbol{i}$, $\boldsymbol{j}$, $\boldsymbol{k}$, is an element of the form
\begin{equation*}
p=a_{0}+a_{1}\boldsymbol{i}+a_{2}\boldsymbol{j}+a_{3}\boldsymbol{k}=\left(
a_{0},a_{1},a_{2},a_{3}\right) ,\text{ }\left( a_{0}\boldsymbol{1}
=a_{0}\right) ,
\end{equation*}
where
\begin{align*}
\boldsymbol{i}^{2} &=\boldsymbol{j}^{2}=\boldsymbol{k}^{2}=-1, \\
\boldsymbol{ij} &=\boldsymbol{k}=-\boldsymbol{ji},\text{ }\boldsymbol{jk}=
\boldsymbol{i}=-\boldsymbol{kj},\text{ }\boldsymbol{ki}=\boldsymbol{j}=-
\boldsymbol{ik}.
\end{align*}

Ram\'{\i}rez \cite{Ram1} defined the $n$th $k$-Fibonacci quaternion, $D_{k,n}$, as
follows:
\begin{equation*}
D_{k,n}=F_{k,n}+F_{k,n+1}\boldsymbol{i}+F_{k,n+2}\boldsymbol{j}+F_{k,n+3}
\boldsymbol{k},\text{ }n\geq 0,
\end{equation*}
where $F_{k,n}$ is the $n$th $k$-Fibonacci number. Ram\'{\i}rez \cite{Ram1} also gave the Binet
formula for the $k$-Fibonacci quaternion as follows:
\begin{equation*}
D_{k,n}=\frac{\widehat{\alpha }\alpha ^{n}-\widehat{\beta }\beta ^{n}}{
\alpha -\beta },
\end{equation*}
where $\widehat{\alpha }=1+\alpha \boldsymbol{i}+\alpha ^{2}\boldsymbol{j}
+\alpha ^{3}\boldsymbol{k}$ and $\widehat{\beta }=1+\beta \boldsymbol{i}
+\beta ^{2}\boldsymbol{j}+\beta ^{3}\boldsymbol{k}$.

Ram\'{\i}rez \cite{Ram1} conjectured that the Catalan identity for the $k$-Fibonacci
quaternions is
\begin{equation*}
D_{k,n-r}D_{k,n+r}-D_{k,n}^{2}=\left( -1\right) ^{n-r}\left(
2F_{k,r}D_{k,r}-G_{k,r}\boldsymbol{k}\right) ,
\end{equation*}
for $n\geq r\geq 1$, where $G_{k,r}$ is the sequence satisfying the following recurrence:
\begin{equation*}
G_{k,0}=0,\text{ }G_{k,1}=k^{2}+2k,\text{ and }G_{k,n}=\left( k^{2}+2\right)
G_{k,n-1}-G_{k,n-2},\text{ }n\geq 2.
\end{equation*}
However, in this short paper, we show that this conjecture is incorrect, by
giving the correct Catalan identity and proving it.

\section{Catalan identity for the $k$-Fibonacci quaternions}

 We need the following lemma.

\begin{lemma}\label{le1}
For $r\geq 1$, we have
\begin{equation*}
\frac{\widehat{\alpha }\widehat{\beta }\beta ^{r}-\widehat{\beta }\widehat{
\alpha }\alpha ^{r}}{\alpha -\beta }=-2D_{k,r}+L_{k,2}L_{k,r}\boldsymbol{k}.
\end{equation*}
\end{lemma}

\begin{proof}
Since
\begin{equation*}
\widehat{\alpha }\widehat{\beta }=2+2\beta \boldsymbol{i}+2\beta ^{2}
\boldsymbol{j}+\left( \alpha ^{3}+\beta ^{3}+\alpha -\beta \right)
\boldsymbol{k}
\end{equation*}
and
\begin{equation*}
\widehat{\beta }\widehat{\alpha }=2+2\alpha \boldsymbol{i}+2\alpha ^{2}
\boldsymbol{j}+\left( \alpha ^{3}+\beta ^{3}+\beta -\alpha \right)
\boldsymbol{k},
\end{equation*}
we get
\begin{align*}
\frac{\widehat{\alpha }\widehat{\beta }\beta ^{r}-\widehat{\beta }\widehat{
\alpha }\alpha ^{r}}{\alpha -\beta } &=-2F_{k,r}-2F_{k,r+1}\boldsymbol{i}
-2F_{k,r+2}\boldsymbol{j}+\left( -2F_{k,r+3}+L_{k,2}L_{k,r}\right)
\boldsymbol{k} \\
&=-2D_{k,r}+L_{k,2}L_{k,r}\boldsymbol{k}.
\end{align*}
\end{proof}

\begin{theorem}
For $n\geq r\geq 1$, Catalan identity for the $k$-Fibonacci quaternions is
\begin{equation*}
D_{k,n-r}D_{k,n+r}-D_{k,n}^{2}=\left( -1\right) ^{n-r+1}\left(
2F_{k,r}D_{k,r}-L_{k,2}F_{k,2r}\boldsymbol{k}\right) .
\end{equation*}
\end{theorem}

\begin{proof}
By considering the Binet formula for the $k$-Fibonacci quaternions,
quaternion multiplication and Lemma \ref{le1}, we obtain
\begin{align*}
D_{k,n-r}D_{k,n+r}-D_{k,n}^{2} &=\left( \frac{\widehat{\alpha }\alpha
^{n-r}-\widehat{\beta }\beta ^{n-r}}{\alpha -\beta }\right) \left( \frac{
\widehat{\alpha }\alpha ^{n+r}-\widehat{\beta }\beta ^{n+r}}{\alpha -\beta }
\right) -\left( \frac{\widehat{\alpha }\alpha ^{n}-\widehat{\beta }\beta ^{n}
}{\alpha -\beta }\right) ^{2} \\
&=\frac{\left( \alpha \beta \right) ^{n}}{\left( \alpha -\beta \right) ^{2}}
\left( \widehat{\alpha }\widehat{\beta }\left( 1-\frac{\beta ^{r}}{\alpha
^{r}}\right) +\widehat{\beta }\widehat{\alpha }\left( 1-\frac{\alpha ^{r}}{
\beta ^{r}}\right) \right)  \\
&=\left( \alpha \beta \right) ^{n}\frac{\alpha ^{r}-\beta ^{r}}{\left(
\alpha -\beta \right) ^{2}}\left( \frac{\widehat{\alpha }\widehat{\beta }}{
\alpha ^{r}}-\frac{\widehat{\beta }\widehat{\alpha }}{\beta ^{r}}\right)  \\
&=\left( \alpha \beta \right) ^{n-r}\frac{\alpha ^{r}-\beta ^{r}}{\alpha
-\beta }\left( \frac{\widehat{\alpha }\widehat{\beta }\beta ^{r}-\widehat{
\beta }\widehat{\alpha }\alpha ^{r}}{\alpha -\beta }\right)  \\
&=\left( \alpha \beta \right) ^{n-r}F_{k,r}\left( -2D_{k,r}+L_{k,2}L_{k,r}
\boldsymbol{k}\right)  \\
&=\left( -1\right) ^{n-r+1}\left( 2F_{k,r}D_{k,r}-L_{k,2}F_{k,2r}
\boldsymbol{k}\right) .
\end{align*}
\end{proof}


\bigskip

\begin{thebibliography}{9}
\bibitem{Ram1} J. L. Ram\'{\i}rez, Some combinatorial properties of the $k$
-Fibonacci and the $k$-Lucas quaternions, {\it An. \c Stiin\c t. Univ. Ovidius Constan\c ta Ser. Mat.} {\bf 23} (2015),
201--212.
\end{thebibliography}

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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B39.

\noindent \emph{Keywords:} $k$-Fibonacci quaternion, Catalan's identity.

\bigskip
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\vspace*{+.1in}
\noindent
Received May 18 2015;
revised version received  May 20 2015; June 18 2015; June 24 2015.
Published in {\it Journal of Integer Sequences}, July 16 2015.

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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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