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\begin{document}

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\begin{center}
\vskip 1cm{\LARGE\bf The Star of David and Other Patterns \\
\vskip .1in
in Hosoya Polynomial Triangles}
\vskip 1cm
\large
Rigoberto Fl\'orez\\
Department of Mathematics and Computer Science\\
The Citadel\\			
Charleston, SC 29409 \\
USA \\
\href{mailto:rigo.florez@citadel.edu}{\tt rigo.florez@citadel.edu} \\
\ \\
Robinson A. Higuita\\
Instituto de Matem\'aticas\\
Universidad de Antioquia\\
Medell\'in\\
Colombia\\
\href{mailto:robinharra@yahoo.es}{\tt robinson.higuita@udea.edu.co}\\
\ \\
Antara Mukherjee\\
Department of Mathematics and Computer Science\\
The Citadel\\
Charleston, SC  29409 \\
USA \\
\href{mailto:antara.mukherjee@citadel.edu}{\tt antara.mukherjee@citadel.edu}

\end{center}

\vskip .2 in

\begin{abstract}

We define two types of second-order polynomial sequences. A sequence is of \emph{Fibonacci-type} (\emph{Lucas-type})
if its Binet formula is similar in structure to the Binet formula for the Fibonacci (Lucas) numbers.
Familiar examples are Fibonacci polynomials, Chebyshev polynomials, Morgan-Voyce
polynomials, Lucas polynomials, Pell polynomials, Fermat polynomials, Jacobsthal polynomials, Vieta
polynomials and other known sequences of polynomials.

We generalize the numerical recurrence relation given by Hosoya to polynomials by constructing a Hosoya
triangle for polynomials where each entry is either a product of two polynomials of Fibonacci-type or
a product of two polynomials of Lucas-type. For every such choice of polynomial sequence we
obtain a triangular array of polynomials. In this paper we extend the star of David property, also called
the Hoggatt-Hansell identity, to these types of triangles. In addition, we study other geometric patterns
in these triangles and as a consequence we obtain geometric interpretations for the Cassini identity,
the Catalan identity, and other identities for Fibonacci polynomials.
\end{abstract}

\section {Introduction}

A second-order polynomial sequence is of \emph{Fibonacci-type} (\emph{Lucas-type}) if its Binet formula
has a structure similar to that for Fibonacci (Lucas) numbers. Familiar examples of such polynomials are
Fibonacci polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials,
Pell polynomials, Fermat polynomials, Jacobsthal polynomials, Vieta polynomials, and other sequences of polynomials.
Most of the polynomials mentioned here are  discussed by Koshy \cite{koshy, koshy1}.

The \emph{Hosoya triangle}, formerly called the Fibonacci triangle,  \cite{florezHiguitaMuk, florezjunes, hosoya, koshy},
consists of a triangular array of numbers where each entry is a product of two Fibonacci numbers (see \seqnum{A058071}).
If in this triangle we replace the Fibonacci numbers with the corresponding polynomials from the sequences mentioned above,
we obtain a Hosoya-like polynomial triangle (see Tables \ref{tabla1} and \ref{tabla_equivalent}).
Therefore, for every choice of a polynomial sequence we obtain a distinct Hosoya polynomial triangle.
So, every polynomial evaluation gives rise to a  numerical triangle (see Table \ref{Tableahosoyatriangles}).
In particular the classic Hosoya triangle can be  obtained by evaluating the entries of Hosoya polynomial
triangle arising from Fibonacci polynomials evaluated at $x=1$. For brevity we call
these triangles Hosoya polynomial triangles and if there is no ambiguity we call them Hosoya triangles.

The Hosoya polynomial triangle provides a good geometric way to study algebraic and combinatorial
properties of products of recursive sequences of polynomials. In this paper we study some of its geometric properties.
Note that any geometric property in this triangle is automatically true for the classic (numerical) Hosoya triangle.

\begin{figure} [!ht]
\begin{center}
\includegraphics[width=33mm]{stard_Hex.eps}
\end{center}
\caption{Star of David from a hexagon. } \label{starofDavidHex}
\end{figure}

A hexagon gives rise to the star of David --- connecting its alternating vertices with a continuous line as in
Figure \ref{starofDavidHex}. Given a hexagon in a Hosoya triangle can one determine whether the vertices of
the two triangles of the star of David have the same greatest common divisor (gcd)? If
both greatest common divisors are equal, then we say that the star of David has the \emph{gcd property}.
Several authors have been interested in this property, see for example
\cite{florezHiguitaJunes,florezjunes,hoggatt_star_david,koshygibonomial,Korntved,Long, sun}.
For instance, in 2014 Fl\'{o}rez et al.~\cite{florezHiguitaJunesGCD} proved the star of
David property in the generalized Hosoya triangle. Koshy \cite{koshygibonomial, koshy1} defined the
gibonomial triangle and proved one of the fundamental properties of the star of David in this triangle.
In a short comment we establish the gcd property of the star of David for the gibonomial triangle.


Since every polynomial sequence of \emph{Fibonacci-type} or of \emph{Lucas-type} gives rise to a Hosoya triangle,
the above question seems complicated to answer. We prove that the star of David property holds for most of the
cases (depending on the locations of its points in the Hosoya triangle). We also prove that if the
star of David property does not hold, then the two gcds are proportional. We give a characterization of the members of
the family of Hosoya triangles that satisfy the star of David property. From Table \ref{equivalent}, we obtain
a sub-family of fourteen distinct Hosoya triangles. We provide a complete classification of the members that
satisfy the star of David property.

We also study other geometric properties that hold in a Hosoya triangle, called the
\emph{rectangle property} and the \emph{zigzag property}. A rectangle in the  Hosoya polynomial
triangle is a set of four points in the triangle that are arranged as the vertices of a rectangle. Using the
rectangle property we give geometric interpretations and proofs of the Cassini, Catalan, and
Johnson identities for \emph{Fibonacci-type} or for \emph{Lucas-type} sequences.

 \section{Preliminaries and the main theorem}\label{Preliminaries}

 In this section we summarize some concepts given by the authors in earlier articles. For example,
 the authors \cite{florezHiguitaMukCharact} have studied the polynomial sequences given here.
 The authors \cite{florezHiguitaMuk} have also studied  polynomial triangular array. Throughout the paper we
 consider polynomials in $\mathbb{Q}[x]$.  The polynomials in the Subsection
 \ref{General:Fibonacci:Polynomial} are presented in a formal way.  However, for brevity and if there is no
 ambiguity after Subsection \ref{General:Fibonacci:Polynomial} and throughout the paper we avoid these
 formalities. Thus, we present the polynomials without explicit use of ``$x$". We return to this
 formality if we need to evaluate a polynomial at a particular value. Another exception of this mentioned
 informality are the familiar examples of Fibonacci-type and Lucas-type  polynomials. We adhere to the conventional
 formality as they appear in the literature (see, for example, Table \ref{equivalent}).

 \subsection{Second-order polynomial sequences}\label{General:Fibonacci:Polynomial}

We now define two types of second-order polynomial recurrence relations:

\begin{equation}\label{Fibonacci;general:FT}
\Ft{0}(x)=0, \; \Ft{1}(x)= 1,\;  \text{and} \;  \Ft{n}(x)= d(x) \Ft{n - 1}(x) + g(x) \Ft{n - 2}(x) \text{ for } n\ge 2,
\end{equation}
where $d(x)$, and $g(x)$ are fixed non-zero polynomials in $\mathbb{Q}[x]$.

We say a polynomial recurrence relation is of \emph{Fibonacci-type} if it satisfies the relation given
in \eqref{Fibonacci;general:FT}, and of \emph{Lucas-type} if
\begin{equation}\label{Fibonacci;general:LT}
\Lt{0}(x)=p_{0}, \; \Lt{1}(x)= p_{1}(x),\;  \text{and} \;  \Lt{n}(x)= d(x) \Lt{n - 1}(x) + g(x) \Lt{n - 2}(x) \text{ for } n\ge 2,
\end{equation}
where $|p_{0}|=1$ or $2$ and $p_{1}(x)$, $d(x)=\alpha p_{1}(x)$, and $g(x)$ are fixed non-zero  polynomials in $\mathbb{Q}[x]$ with
$\alpha$ an integer of the form $2/p_{0}$.

To use similar notation for \eqref{Fibonacci;general:FT} and \eqref{Fibonacci;general:LT} on certain occasions we write
$p_{0}=0$, $p_{1}(x)=1$ to indicate the initial conditions of Fibonacci-type polynomials. Some familiar examples
of Fibonacci-type polynomials and of Lucas-type polynomials are in Table \ref{equivalent}
(see also \cite{florezHiguitaMukCharact, florezHiguitaMuk, Pell,Fermat,koshy}).

If $G_{n}$ is either $\Ft{n}$ or $\Lt{n}$ for all $n\ge 0$ and $d^2(x)+4g(x)&gt; 0$ then the explicit formula for the
recurrence relations in \eqref{Fibonacci;general:FT} and \eqref{Fibonacci;general:LT}  is given by
\begin{equation*}\label{solutionrecurrencerelationuno}
 G_{n}(x) = t_1 a^{n}(x) + t_2 b^{n}(x),
\end{equation*}
where $a(x)$ and $b(x)$ are the solutions of the quadratic equation associated to the second-order
recurrence relation $G_{n}(x)$. That is,  $a(x)$ and $b(x)$ are the solutions of $z^2-d(x)z-g(x)=0$.
If $\alpha=2/p_{0}$, then the Binet formula for Fibonacci-type polynomials is stated
in  \eqref{bineformulauno} and the Binet formula for Lucas-type polynomials is stated in \eqref{bineformulados}
(for details on the construction of the two Binet formulas see \cite{florezHiguitaMukCharact}).
\begin{equation}\label{bineformulauno}
\Ft{n}(x) = \dfrac{a^{n}(x)-b^{n}(x)}{a(x)-b(x)}
\end{equation}
and
\begin{equation}\label{bineformulados}
\Lt{n}(x)=\dfrac{a^{n}(x)+b^{n}(x)}{\alpha}.
\end{equation}

Note that for both types of sequences the identities
$$a(x)+b(x)=d(x), \quad a(x)b(x)= -g(x), \quad \text{ and } \quad a(x)-b(x)=\sqrt{d^2(x)+4g(x)}$$
hold,
where $d(x)$ and $g(x)$ are the polynomials defined in \eqref{Fibonacci;general:FT} and \eqref{Fibonacci;general:LT}.

A sequence of Lucas-type (Fibonacci-type) is \emph{equivalent} or \emph{conjugate} to a sequence of Fibonacci-type (Lucas-type),
if their recursive sequences are determined by the same polynomials $d(x)$ and $g(x)$. Notice that two equivalent polynomials
have the same $a(x)$ and $b(x)$ in their Binet representations. Examples of equivalent polynomials are given in Table \ref{equivalent}.
Note that the leftmost polynomials in Table \ref{equivalent} are of Lucas-type and their equivalent Fibonacci-type polynomials
are in the fifth column on the same line.

 For most of the proofs involving these sequences it is required that
 \begin{equation}\label{extra:contion}
 \gcd(p_{0}, p_{1}(x))=1, \quad \gcd(p_{0}, d(x))=1, \quad \gcd(p_{0}, g(x))=1,  \text{ and }  \gcd(d(x), g(x))=1.
 \end{equation}
Therefore, for the rest the paper we suppose that these four conditions hold for both types of sequence studied here.
We use $\rho$ to denote $\gcd(d(x),G_{1}(x))$. Notice that in the definition of Pell-Lucas we have
$Q_{0}(x)=2$ and $Q_{1}(x)=2x$. Thus, the $\gcd(2,2x)=2\ne 1$.
Therefore, Pell-Lucas does not satisfy the extra conditions that we imposed in \eqref{extra:contion}. So,
to resolve this inconsistency we use $Q^{\prime}_{n}(x)=Q_{n}(x)/2$ instead of $Q_{n}(x)$.
Fl\'orez et al.~\cite{florezHiguitaJunes} have worked on similar problems for numerical sequences.

\begin{table} [!ht]
\begin{center}\scalebox{0.8}{
\begin{tabular}{|l|c|l|r||lc|l|c|} \hline
   Polynomial of    &amp; $\Lt{n}(x)$ 	     &amp; $d(x )$ &amp;$g(x)$&amp;Polynomial of 	&amp;$\Ft{n}(x)$   	&amp; $d(x)$ &amp;$g(x)$\\	
   Lucas-type 	 	&amp; 		     		&amp;	      &amp;	 	  &amp; Fibonacci-type  &amp;			    &amp;        &amp;		\\ \hline \hline
   Lucas 			&amp;$D_{n}(x)$  		&amp;$x$     &amp;1       &amp;Fibonacci 		&amp;$F_{n}(x)$ 	&amp; $x$    &amp; $1$	\\ 						
   Pell-Lucas 		&amp;$Q_{n}(x)$  		&amp; $2x$   &amp; $1$	  &amp;Pell		        &amp; $P_{n}(x)$    &amp; $2x$   &amp; $1$  \\
   Fermat-Lucas 	&amp; $\vartheta_{n}(x)$&amp; $x$    &amp; $-2$   &amp; Fermat 			&amp; $\Phi_{n}(x)$ &amp; $x$    &amp; $-2$ \\
   Chebyshev first kind&amp; $T_{n}(x)$ 	&amp; $2x$   &amp; $-1$   &amp;Chebyshev second kind&amp;$U_{n}(x)$ &amp; $2x$   &amp; $-1$ \\
   Jacobsthal-Lucas	&amp; $j_{n}(x)$	    &amp; 1      &amp;$2x$    &amp;Jacobsthal  		&amp; $J_{n}(x)$    &amp;1       &amp; $2x$ \\
   Morgan-Voyce 	&amp;$C_{n}(x)$ 	   	&amp;$x+2$   &amp; $-1$   &amp;Morgan-Voyce	    &amp; $B_{n}(x)$ 	&amp; $x+2$  &amp; $-1$\\
   Vieta-Lucas 		&amp;$v_{n}(x)$ 	   	&amp; $x$    &amp; $-1$   &amp;Vieta	        &amp; $V_{n}(x)$ 	&amp; $x$    &amp; $-1$ \\   \hline
\end{tabular}}
\end{center}
\caption{$\Ft{n}$ equivalent to $\Lt{n}$.} \label{equivalent}
\end{table}

\subsection{Hosoya polynomial triangle} \label{HosoyaSection}

We now give a precise definition of both the Hosoya polynomial sequence and the Hosoya polynomial triangle.
We recall that for brevity throughout the paper we present  the polynomials 
without specifying the variable ``$x$".
For example, instead of $\Ft{n}(x)$ we use $\Ft{n}$.

Let $p_{0}$, $p_{1}$, $d$,  and $g$ be fixed polynomials as defined in
\eqref{Fibonacci;general:FT} and \eqref{Fibonacci;general:LT}. Then the \emph{Hosoya polynomial} sequence
$\left\{H(r,k)\right\}_{r,k\ge 0}$ is defined using the double recursion
\[ H(r,k)= d  H(r-1,k)+g  H(r-2,k)\]
 and
 \[ H(r,k)= d  H(r-1,k-1)+g  H(r-2,k-2) ,\]
where $ r&gt;1 $ and $0\le k \le r-1$, with initial conditions
\[H(0,0)=p_{0}^2; \quad H(1,0)=p_{0} p_{1} ; \quad H(1,1)=p_{0} p_{1} ; \quad H(2,1)=p_{1}^2.\]
This sequence gives rise to the \emph{Hosoya polynomial triangle}, where the
entry in position $k$ (taken from left to right), of the $r${th} row is equal to $H(r,k)$ (see Table \ref{tabla1}).

\begin{table} [!ht] \small
\begin{center} \addtolength{\tabcolsep}{-3pt} \scalebox{.9}{
\begin{tabular}{ccccccccccc}
&amp;&amp;&amp;&amp;&amp;                                                    $H(0,0)$                                                 &amp;&amp;&amp;&amp;&amp;\\
&amp;&amp;&amp;&amp;                                        $H(1,0)$     &amp;&amp;     $H(1,1)$                                           &amp;&amp;&amp;&amp;\\
&amp;&amp;&amp;                                $H(2,0)$    &amp;&amp;     $H(2,1)$     &amp;&amp;     $H(2,2)$                                  &amp;&amp;&amp;\\
&amp;&amp;                        $H(3,0)$   &amp;&amp;     $H(3,1)$     &amp;&amp;     $H(3,2)$      &amp;&amp;    $H(3,3)$                        &amp;&amp;\\
&amp;              $H(4,0)$     &amp;&amp;     $H(4,1)$    &amp;&amp;     $H(4,2)$     &amp;&amp;     $H(4,3)$     &amp;&amp;     $H(4,4)$               &amp;\\
      $H(5,0)$     &amp;&amp;    $H(5,1)$    &amp;&amp;     $H(5,2)$     &amp;&amp;     $H(5,3)$      &amp;&amp;    $H(5,4)$     &amp;&amp;    $H(5,5)$     \\
\end{tabular}}
\end{center}
\caption{Hosoya polynomial triangle.} \label{tabla1}
\end{table}


We say that the Hosoya triangle is of Fibonacci-type, denoted $H_{\Ft{}}$,  if $p_{0}=0$ and $p_{1}=1$,
and $d$  and $g$ are as in  \eqref{Fibonacci;general:FT}. Similarly, the Hosoya triangle of Lucas-type
(denoted $H_{\Lt{}}$) can be defined.

In the definition of the Hosoya polynomial sequence the polynomials $d$, $g $, $p_{0}$, and $p_{1} $ can be any four polynomials
in $\mathbb{Q}[x]$. Thus, these four polynomials need not be as defined in \eqref{Fibonacci;general:FT} and \eqref{Fibonacci;general:LT}.
However, in this paper we impose the restrictions above since we want a relationship between the sequences of Fibonacci-type and of  Lucas-type
for the Hosoya polynomial triangles. This relation is given by Proposition \ref{lemma0} (see Fl\'orez et al.~\cite{florezHiguitaMuk, florezHiguitaJunesGCD}).

\begin{proposition}[\cite{florezHiguitaMuk}]\label{lemma0}
If $G_n$ is either $\Ft{n}$ or $\Lt{n}$ for all $n\ge 0$, then $H(r,k)= G_k G_{r-k}$.
\end{proposition}

Proposition \ref{lemma0} implies that the entries of a Hosoya polynomial triangle are the product of two polynomials
that are of the form as described in \eqref{Fibonacci;general:FT} or in \eqref{Fibonacci;general:LT}.
We observe that Table \ref{tabla1} together with this proposition give rise to Table \ref{tabla_equivalent}.

\begin{table} [!ht] \small
\begin{center} \addtolength{\tabcolsep}{-1pt} \scalebox{.9}{
\begin{tabular}{ccccccccccc}
&amp;&amp;&amp;&amp;&amp;           			  	        $G_{0} \, G_{0}$                             				                        &amp;&amp;&amp;&amp;&amp;\\
&amp;&amp;&amp;&amp;         	      		  $G_{0}\, G_{1}$ &amp;&amp;      $G_{1} \,G_{0}$                                           		   	  &amp;&amp;&amp;&amp;\\
&amp;&amp;&amp;             	    $G_{0}\,G_{2}$   &amp;&amp;     $G_{1}\,G_{1}$     &amp;&amp;  $G_{2}\,G_{0}$                                          &amp;&amp;&amp;\\
&amp;&amp;             $G_{0}\,G_{3}$   &amp;&amp;     $G_{1}\,G_{2}$     &amp;&amp;  $G_{2}\,G_{1}$  &amp;&amp;   $G_{3}\,G_{0}$                     	        &amp;&amp;\\
&amp;     $G_{0}\,G_{4}$    &amp;&amp;    $G_{1}\,G_{3}$     &amp;&amp;   $G_{2}\,G_{2}$  &amp;&amp;   $G_{3}\,G_{1}$   &amp;&amp;  $G_{4}\,G_{0}$                    &amp;\\
 $G_{0}\,G_{5}$   &amp;&amp;   $G_{1}\,G_{4}$      &amp;&amp;   $G_{2}\,G_{3}$   &amp;&amp;   $G_{3}\,G_{2}$   &amp;&amp;   $G_{4}\,G_{1}$   &amp;&amp;   $G_{5}\,G_{0}$   \\
\end{tabular}}
\end{center}
\caption{$H(r,k)= G_k G_{r-k} $.} \label{tabla_equivalent}
\end{table}

Some examples of $H(r,k)$ are in Table \ref{polynomialformula}, obtained from Table \ref{equivalent} using Proposition \ref{lemma0}.
Therefore, some examples of Hosoya polynomial triangles can be constructed using Tables \ref{tabla_equivalent} and \ref{polynomialformula}.
It is enough to substitute each entry in Table \ref{tabla1} or Table \ref{tabla_equivalent} by the corresponding entry in
Table \ref{polynomialformula}. Thus, we obtain a Hosoya polynomial triangle for each of the specific polynomials mentioned
in Table \ref{equivalent}. So, Table \ref{polynomialformula} gives rise to 14 examples of Hosoya polynomial triangles.

For example, using the first polynomial in  Table \ref{polynomialformula} and Proposition \ref{lemma0} in
Table \ref{tabla_equivalent} we obtain the Hosoya polynomial triangle $H_{F}$ where the entry $H(r,k)$ is equal to
$F_{k}(x) F_{r-k}(x)$. This is represented in Table \ref{tabla2} without the points that contain the factor $F_{0}(x)=0$.

\begin{table} [!ht]
\begin{center}\scalebox{0.8}{
\begin{tabular}{|l|c|c|c|r||l|c|c|c|r|} \hline
$H(r,k)$ &amp; $p_{0}$ &amp; $p_{1}$ &amp; $d $ &amp; $g  $ &amp; $H(r,k)$&amp; $p_{0}$ &amp; $p_{1}$ &amp; $d $ &amp; $g $
\\ \hline \hline \noalign {\smallskip}
    $F_k(x)F_{r-k}(x)$      &amp; 0 &amp; $1$  &amp; $x$ &amp; $1$ &amp;
    $D_k(x)D_{r-k}(x)$      &amp; 2 &amp; $2x$ &amp; $x$ &amp; $1$\\
    $P_k(x)P_{r-k}(x)$      &amp; 0 &amp; $1$ &amp; $2x$ &amp; $1$&amp;
    $Q_k(x)Q_{r-k}(x)$      &amp; 2 &amp; $2x$ &amp; $2x$ &amp; $1$\\
    $\Phi_k(x)\Phi_{r-k}(x)$&amp; 0 &amp; $1$ &amp; $x$ &amp; $-2$&amp;
    $\vartheta_k(x)\vartheta_{r-k}(x)$ &amp; $2$ &amp; $3x$ &amp; $x$ &amp; $-2$\\
    $U_k(x)U_{r-k}(x)$      &amp; 0 &amp; $1$ &amp; $2x$ &amp; $-1$&amp;
    $T_k(x)T_{r-k}(x)$      &amp; 1 &amp; $x$ &amp; $2x$ &amp; $-1$\\
    $J_k(x)J_{r-k}(x)$      &amp; 0 &amp; 1 &amp;1 &amp; $2x$&amp;
    $j_k(x)j_{r-k}(x)$      &amp; 2 &amp; 1 &amp; 1 &amp;$2x$ \\
    $B_k(x)B_{r-k}(x)$      &amp; 0 &amp; $1$ &amp; $x+2$ &amp; $-1$&amp;
    $C_k(x)C_{r-k}(x)$      &amp; 2 &amp; $x+2$ &amp; $x+2$ &amp; $-1$\\
    $V_k(x)V_{r-k}(x)$      &amp; 0 &amp; $1$ &amp; $x$ &amp; $-1$ &amp;
    $v_k(x)v_{r-k}(x)$      &amp; 2 &amp; $x$ &amp; $x$ &amp; $-1$\\
     \hline
\end{tabular}}
\end{center}
\caption{Terms $H(r,k)$ of the Hosoya polynomial triangle.} \label{polynomialformula}
\end{table}

Observe that $H(r,k)$ in the first column of Table \ref{polynomialformula} is a product of polynomials of Fibonacci-type.
Therefore, $G_{0} =0$. So, the edges containing $G_{0} $ as a factor in Table \ref{tabla_equivalent}, will have entries equal
to zero. From the sixth column of Table \ref{polynomialformula} we see that $H(r,k)$ is a product of polynomials of Lucas-type.
So, the edges containing $G_{0} $ as a factor in Table \ref{tabla_equivalent} will not have entries equal to zero.

\begin{table} [!ht] \small
\begin{center} \addtolength{\tabcolsep}{-3pt} \scalebox{.9}{
\begin{tabular}{ccccccccccc}
&amp;&amp;&amp;&amp;&amp;                                 $1$                                          &amp;&amp;&amp;&amp;&amp;\\
&amp;&amp;&amp;&amp;                           $x$     &amp;&amp;       $x$                    			    &amp;&amp;&amp;&amp;\\
&amp;&amp;&amp;                $x^2+1$     &amp;&amp;     $x^2$      &amp;&amp;  $x^2+1$             	         &amp;&amp;&amp;\\
&amp;&amp;          $x^3+2x$   &amp;&amp;     $x(x^2+1)$   &amp;&amp;     $x(x^2+1)$  &amp;&amp;     $x^3+2x$         &amp;&amp;\\
&amp; $x^4+3x^2+1$ &amp;&amp;  $x(x^3+2x)$  &amp;&amp;  $(x^2+1)^2$ &amp;&amp;   $x(x^3+2x)$  &amp;&amp;    $x^4+3x^2+1$    &amp;\\
\end{tabular}}
\end{center}
\caption{The Hosoya triangle $H_{F}$ where $H(r,k) = F_{k}(x) F_{r-k}(x)$.} \label{tabla2}
\end{table}


\subsection{Star of David property in the Hosoya polynomial triangle}

In this subsection we state the main results, namely the star of David properties for both type
Hosoya polynomial triangle, Lucas-type and Fibonacci-type. These properties hold in the Pascal triangle,
the Fibonomial triangle, the gibonomial triangle, and in both the Hosoya and the generalized Hosoya triangle.

Koshy \cite[Chapters 6 and 26]{koshytriangular} discussed that some properties of star of  David are present in
several triangular arrays. These properties --- called the \emph{Hoggatt-Hansell} identity,
the \emph{Gould} property, or \emph{gcd} property --- were also proved in \cite{florezHiguitaJunesGCD, florezjunes} for Hosoya
and generalized Hosoya triangles. The results in this paper generalize several results in the articles
\cite{florezHiguitaJunesGCD, florezjunes, hosoya, koshytriangular} that were proved for numerical sequences.

Those familiar with the gibonomial triangle (see Koshy \cite{koshygibonomial} or Sagan \cite{Sagan}), may find
it interesting that the gcd property also holds in this triangle.  The proof of this fact follows by adapting
the numerical proof in Hillman and Hoggatt \cite{hoggattHillman}, to polynomials.

From Figure \ref{starofDavidHosoya} we can see that the star of David is formed by two triangles. For the rest of paper
when we refer to the star of David we assume that it is embedded in a Hosoya polynomial triangle. We show that the product
of points in one triangle equals the product of points in the second triangle. We also find
conditions that ensure that the gcd of the points in the leftmost triangle are equal to the gcd of the points in the rightmost triangle
(this is true if $\gcd(\rho, G_{n} /\rho )=1$, where  $\rho=\gcd(d ,G_{1} )$ and $G_n$ is either $\Ft{n}$ or $\Lt{n}$).
For example,  the polynomials in Table \ref{equivalent} that satisfy this condition are: Fibonacci, Lucas,
Pell-Lucas, Chebyshev first kind, Jacobsthal, Jacobsthal-Lucas, and both
Morgan-Voyce polynomials. The polynomials in Table \ref{equivalent} that satisfy $\gcd(\rho^2, G_{n} )\not=1$ are:
Pell, Fermat, Fermat-Lucas, and Chebyshev second kind.

\begin{figure} [!ht]
\begin{center}
\includegraphics[width=100mm]{stard_Hosoya_triangle.eps}
\end{center}
\caption{Star of David in a Hosoya triangle where $G_k$ is either $\Ft{k}$ or $\Lt{k}$ for all $k\ge 0$. } \label{starofDavidHosoya}
\end{figure}

In the following three theorems we generalize the \emph{Hoggatt-Hansell} identity  and \emph{Gould}  property to polynomials.
We  also analyze the relationship between the point that is within the two triangles  of the star of David (see the point $c$ in
Figure \ref{starofDavidHosoya}) and the two diagonals of the star of David. We now state the main results  --- for their proofs see
Section \ref{Divisibility:Properties:Main:Thm} page \pageref{ProofsMainTheorems}.
We recall that for brevity we always suppose that the star of David is embedded in a Hosoya polynomial triangle.

\begin{theorem} \label{gcdstarofdavid:Part:2} Suppose that $\Ft{m+1}\Ft{n-2}$, $\Ft{m}\Ft{n}$, and $\Ft{m+2}\Ft{n-1}$
are the points in a triangle of the star of David and $\Ft{m}\Ft{n-1}$,  $\Ft{m+2}\Ft{n-2}$, and  $\Ft{m+1}\Ft{n}$
are the points in the second triangle  of the star of David. If  $m\ge 1$ and $n&gt;1$, then

\begin{enumerate}[(1)]
\item
$\gcd( \Ft{m+1} \Ft{n-2},\Ft{m} \Ft{n},\Ft{m+2} \Ft{n-1})$ is equal to
 \[
 \begin{cases}
         \beta \gcd(\Ft{m} \Ft{n-1}, \Ft{m+2} \Ft{n-2} , \Ft{m+1} \Ft{n} ), &amp; \mbox{if $m$ and $n$ are both even;} \\
         \gcd(\Ft{m} \Ft{n-1}, \Ft{m+2} \Ft{n-2} , \Ft{m+1} \Ft{n} ), &amp; \mbox{otherwise,}
\end{cases}
 \]
where $\beta$ is a constant that depends on $d , m$, and $n$.

\item Let $c=\Ft{m+1} \Ft{n-1} $ be the point within the two triangles of the star of David.
Then $\gcd(\Ft{m+1} \Ft{n-2}, \Ft{m+1} \Ft{n})\cdot \gcd(\Ft{m} \Ft{n-1} ,\Ft{m+2} \Ft{n-1} )$
is equal to either $c$, $c \Ft{2}$,  or  $c \Ft{2}^2$.
\end{enumerate}
\end{theorem}

\begin{theorem}\label{gcdstarofdavid:Part:3} Suppose that $\Lt{m+1}\Lt{n-2}$, $\Lt{m}\Lt{n}$, and $\Lt{m+2}\Lt{n-1}$
are the points in a triangle of the star of David and $\Lt{m}\Lt{n-1}$,  $\Lt{m+2}\Lt{n-2}$, and  $\Lt{m+1}\Lt{n}$
are the points in the second triangle of the star of David.
If $m\ge 0$ and $n\ge 0$ and $\Lt{m}\Lt{n} \not = \Lt{0}\Lt{0}$, then

\begin{enumerate}[(1)]
\item
$\gcd( \Lt{m+1} \Lt{n-2},\Lt{m} \Lt{n},\Lt{m+2} \Lt{n-1})$ is equal to
 \[
 \begin{cases}
         \beta^{\prime}  \gcd(\Lt{m} \Lt{n-1}, \Lt{m+2} \Lt{n-2} , \Lt{m+1} \Lt{n} ), &amp; \mbox{if $m$ and $n$ are both even;} \\
         \gcd(\Lt{m} \Lt{n-1}, \Lt{m+2} \Lt{n-2} , \Lt{m+1} \Lt{n} ), &amp; \mbox{otherwise,}
\end{cases}
 \]
where $\beta^{\prime}$ is a constant that depends on $\Lt{1} , m$, and $n$.

\item Let $c= \Lt{m+1}  \Lt{n-1} $ be the point within the two triangles of the star of David.
Then $\gcd( \Lt{m+1}  \Lt{n-2},  \Lt{m+1}  \Lt{n})\cdot \gcd( \Lt{m}  \Lt{n-1} , \Lt{m+2}  \Lt{n-1} )$
is equal to either $c$, $c  \Lt{1}$,  or  $c  \Lt{1}^2$.
\end{enumerate}
\end{theorem}

\begin{theorem}\label{gcdstarofdavid:Part1} Suppose that $G_k$ is either $\Ft{k}$ or $\Lt{k}$ for all $k\ge 0$.
If $G_{m+1} G_{n-2} $, $G_{m} G_{n} $, and $G_{m+2} G_{n-1} $ are the points in a triangle of the star of David and
$G_{m} G_{n-1} $, $G_{m+2} G_{n-2} $, and $G_{m+1} G_{n} $ are the points in the second triangle of the star
of David, where $G_{m} \, G_{n} \not =  G_{0}  \, G_{0} $, then
 \[(G_{m+1} G_{n-2})\cdot (G_{m} G_{n})\cdot (G_{m+2} G_{n-1}) =(G_{m} G_{n-1})\cdot(G_{m+2} G_{n-2})\cdot(G_{m+1} G_{n}) .\]
\end{theorem}

\section{Proof of the main theorems} \label{Divisibility:Properties:Main:Thm}

In this section we prove Theorems \ref{gcdstarofdavid:Part:2}  and \ref{gcdstarofdavid:Part:3}.
The proof of Theorem \ref{gcdstarofdavid:Part1} is straightforward. In addition, we present some
corollaries of the main theorems, a few divisibility properties, and gcd properties that are true
for both types of polynomial sequences. Proposition \ref{prop2;1} is a generalization of
\cite[Proposition 2.2]{florezjunes}, both proofs are similar.

\begin{proposition}\label{prop2;1} Let $a, b, c$ and $d$ be polynomials in $\mathbb{Q}[x]$.
\begin{enumerate}[(1)]
\item If $\gcd(a,b)=1$ and $\gcd(c,d)=1$, then
\[\gcd(ab,cd)= \gcd(a,c)\cdot \gcd(a,d)\cdot \gcd(b,c) \cdot \gcd(b,d).\]

\item  If $\gcd(a,c)=\gcd(b,d)=1$, then
$\gcd(a b,cd)=\gcd(a,d) \cdot \gcd(b,c)$.
\end{enumerate}
\end{proposition}


\begin{proof} The proof of Part  (1) follows from the
multiplication property of the gcd.
The proof of Part (2) follows from \cite[Proposition 2.2]{florezjunes} by replacing $a, b, c$ and $d$
integers by $a, b, c$ and $d$ polynomials in $\mathbb{Q}[x]$.
\end{proof}

\begin{proposition}\label{modulo:dx} If $G_i$ is either $\Ft{i}$ or $\Lt{i}$ for all $i\ge 0$, then
 \[
 G_{m}  \bmod d^2 =
 \begin{cases}
         g^{k-1} \left(kd G_{1} +g G_{0} \right), &amp; \mbox{if $m=2k$;} \\
         g^{k} \left(kd G_{0} +G_{1} \right), &amp; \mbox{if $m=2k+1$.}
\end{cases}
 \]
\end{proposition}

\begin{proof} We use mathematical induction. Let $S(m)$ be the statement
 \[
 G_{m}  \bmod d^2  =
 \begin{cases}
         g^{t-1} \left(td G_{1} +g G_{0} \right), &amp; \mbox{if $m=2t$;} \\
         g^{t} \left(td G_{0} +G_{1} \right), &amp; \mbox{if $m=2t+1$.}
\end{cases}
 \]

The basis step, $S(1)$ and $S(2)$, follows from the following two facts;
\[ G_{1} \equiv G_{1} =g^{0} \left(0 d G_{0} +G_{1} \right) \bmod d^2 \]
and
\[G_{2} \equiv G_{2} =g^{0} \left(d G_{1} +g G_{0} \right) \bmod d^2 .\]

We suppose that $S(m)$ is true for $m=2k$ and $m=2k+1$.
The proof of $S(m+1)$ requires two cases, we prove the case for $m+1=2k+2$, the case $m+1=2k+3$ is similar and is omitted.
We know that $G_{m+1} =d G_{m} +g G_{m-1}$. Thus,
$G_{2k+2} =d G_{2k+1} +g G_{2k}$. This and the inductive hypothesis imply that $G_{2k+2}  \bmod d^2 $ is
\[ d  \left(g^{k} \left(kd G_{0} +G_{1} \right)\right)+g \left(g^{k-1} \left(kd G_{1} +g G_{0} \right) \right). \]
Simplifying, we obtain
 \[G_{2(k+1)}  \equiv \modd{g^{k} \left((k+1)d G_{1} +g G_{0} \right)} {d^2} .\]
This completes the proof.
\end{proof}


\begin{lemma}[\cite{florezHiguitaMukCharact}] \label{gcddistance1;2} If $m$ and $n$ are positive integers, $\Ft{t}$
is a Fibonacci-type polynomial, and $\Lt{t} $ is a Lucas-type polynomial, then these hold

\begin{enumerate}[(1)]	
	 \item   $\gcd(d , \Ft{2n+1} )=\Ft{1}$ and $\gcd(d , \Lt{2n+1} )=\Lt{1}$.
	
	\item  $\gcd(d, \Ft{2n} )= d$ and $\gcd(d , \Lt{2n} )= 1$.
	
	\item $\gcd(g , \Ft{n} )=\gcd(g , \Ft{1} )=1$  and $\gcd(g , \Lt{n} )=\gcd(g , \Lt{1} )=1$.
	
 \item If  $0&lt;|m-n|\le 2$, then
 \[
 \gcd(\Lt{m} ,\Lt{n} )=
 \begin{cases}
         \alpha^{-1}d , &amp; \mbox{if $m$ and $n$ are both odd;} \\
         1, &amp; \mbox{otherwise. }
\end{cases}
 \]

   \item If $0&lt;|m-n|\le 2$, then
 \[
 \gcd(\Ft{m} ,\Ft{n} )=
 \begin{cases}
        d, &amp; \mbox{if $m$ and $n$ are both even;} \\
         1, &amp; \mbox{otherwise. }
\end{cases}
 \]
\end{enumerate}
\end{lemma}

\begin{lemma}\label{lem:Bts} Suppose that $G_k$ is either $\Ft{k}$ or $\Lt{k}$ for all $k\ge 0$.
Let $G_{m+1} G_{n-2} $, $G_{m} G_{n} $, and $G_{m+2} G_{n-1} $ be the points in a triangle of the star of David and
$G_{m} G_{n-1} $, $G_{m+2} G_{n-2} $, and $G_{m+1} G_{n} $ be the points in the second triangle of the star
of David, with $m$ and $n$ positive integers where $G_{m} \, G_{n} \not =  G_{0}  \, G_{0}$. If $\Delta_t=\gcd(G_{t} ,G_{t-2} )$, then
	\[
	\gcd(G_{m} G_{n-1},G_{m+1} G_{n},G_{m+2} G_{n-2})=\gcd(G_{n} ,G_{m} , \Delta_m  \Delta_n )
	\]
	and
	\[\gcd(G_{m+1} G_{n-2},G_{m} G_{n} ,G_{m+2} G_{n-1} )=\gcd(G_{n-2} ,G_{m+2} , \Delta_n  \Delta_m ).\]
	
\end{lemma}

\begin{proof} We prove that
	\[
	\gcd(G_{m} G_{n-1},G_{m+1} G_{n},G_{m+2} G_{n-2})=\gcd(G_{n} ,G_{m} , \Delta_m  \Delta_n ).
	\]
	From Lemma \ref{prop2;1} Part (2) we have
	\[\gcd(G_{m} G_{n-1} ,G_{m+1} G_{n} )=\gcd(G_{m} ,G_{n} )\cdot \gcd(G_{n-1} G_{m+1} ).\]
	Therefore,
	\begin{eqnarray*}
		\gcd\left(G_{m} G_{n-1},G_{m+1} G_{n},G_{m+2} G_{n-2}\right)&amp;=&amp;\gcd\left(\gcd\left(G_{m} G_{n-1} ,G_{m+1} G_{n} \right),G_{m+2} G_{n-2} \right)\\
		&amp;=&amp;\gcd\left( \left(\gcd(G_{m} ,G_{n} )\cdot \gcd\left(G_{n-1} G_{m+1} \right) \right),G_{m+2} G_{n-2} \right).
	\end{eqnarray*}	
	From Lemma \ref{gcddistance1;2} Parts (4) and (5) we know that
	\[\gcd(G_{m+2} G_{n-2} ,\gcd(G_{n-1} ,G_{m+1} ))=1.\]
	So, 	
	\begin{eqnarray*}
	\gcd\left(G_{m} G_{n-1},G_{m+1} G_{n},G_{m+2} G_{n-2}\right)&amp;=&amp;\gcd\left(\gcd\left(G_{m} ,G_{n} \right),G_{m+2} ,G_{n-2} \right)\nonumber\\
	&amp;=&amp;\gcd\left(G_{m} ,G_{n} ,G_{m+2} G_{n-2} \right)\nonumber\\
	&amp;=&amp;\gcd(G_{m} ,\gcd(G_{n} ,G_{m+2} G_{n-2} )).
	\end{eqnarray*}
	This and Lemma \ref{prop2;1} imply that
	\begin{eqnarray}\label{trianglebb:cases}
	\gcd(G_{m} G_{n-1},G_{m+1} G_{n},G_{m+2} G_{n-2})
	&amp;=&amp;\gcd(G_{m} ,\gcd(G_{n} ,G_{m+2}  \Delta_n ))\nonumber\\
	&amp;=&amp;\gcd(G_{n} ,\gcd(G_{m} ,G_{m+2}  \Delta_n ))\nonumber\\
	&amp;=&amp;\gcd(G_{n} ,\gcd(G_{m} , \Delta_m  \Delta_n ))\nonumber\\
	&amp;=&amp;\gcd(G_{n} ,G_{m} , \Delta_m  \Delta_n ).\nonumber
	\end{eqnarray}
	Similarly, we have
$\gcd(G_{m+1} G_{n-2},G_{m} G_{n} ,G_{m+2} G_{n-1} )=\gcd(G_{n-2} ,G_{m+2} , \Delta_n  \Delta_m )$.	
\end{proof}

\subsection{Proof of the main theorems} \label{ProofsMainTheorems}

\begin{proof}[Proof of Theorem \ref{gcdstarofdavid:Part:2}.]
If in Lemma \ref{lem:Bts} we consider $G_{n} =\Ft{n} $, we have
\begin{equation} \label{Equation:Proof:Thm2:1}
\gcd(\Ft{m}\Ft{n-1},\Ft{m+1}\Ft{n},\Ft{m+2}\Ft{n-2} )=\gcd(\Ft{n} ,\Ft{m}, \Delta_m  \Delta_n )
\end{equation}
and
\begin{equation} \label{Equat:Proof:Thm2:2}
\gcd(\Ft{m+1}\Ft{n-2},\Ft{m}\Ft{n} , \Ft{m+2}\Ft{n-1})=\gcd(\Ft{n-2},\Ft{m+2}, \Delta_m  \Delta_n ),
\end{equation}
where $\Delta_t=\gcd(\Ft{t} ,\Ft{t-2})$.

For this proof we consider three cases depending on the parity of $m$ and $n$.

\medskip

\noindent {\bf Case $m$ and $n$ are odd}. From Lemma \ref{gcddistance1;2} Part (5)  we  have $ \Delta_{m}= \Delta_{n}=1$.
This,  \eqref{Equation:Proof:Thm2:1}, and \eqref{Equat:Proof:Thm2:2} imply that
\[
\gcd(\Ft{m}\Ft{n-1},\Ft{m+1}\Ft{n},\Ft{m+2}\Ft{n-2} )=1
\]
and
\[\gcd(\Ft{m+1}\Ft{n-2},\Ft{m}\Ft{n} , \Ft{m+2}\Ft{n-1})=1.\]

\medskip

\noindent{\bf Case $m$ and $n$ have different parity}. From Lemma \ref{gcddistance1;2} Part (5) we have  $ \Delta_{m}  \Delta_{n} =d$.
This,  \eqref{Equation:Proof:Thm2:1},  and \eqref{Equat:Proof:Thm2:2} imply that
\[\gcd(\Ft{m}\Ft{n-1},\Ft{m+1}\Ft{n},\Ft{m+2}\Ft{n-2} )=\gcd(\Ft{n} ,\Ft{m} ,d )\] and
\[\gcd(\Ft{m+1}\Ft{n-2},\Ft{m}\Ft{n} , \Ft{m+2}\Ft{n-1})=\gcd(\Ft{n-2},\Ft{m+2},d).\]
From Lemma \ref{gcddistance1;2} Part (1) we have  $\gcd(\Ft{n} ,\Ft{m} ,d )=1=\gcd(\Ft{n-2},\Ft{m+2},d)$.
Therefore,
$\gcd(\Ft{m}\Ft{n-1},\Ft{m+1}\Ft{n},\Ft{m+2}\Ft{n-2} )=\gcd(\Ft{m+1}\Ft{n-2},\Ft{m}\Ft{n} , \Ft{m+2}\Ft{n-1})=1$.

\medskip

\noindent{\bf Case both $m$ and $n$ are even}. Suppose that $n=2k_{1}$ and $m=2k_{2}$ for some $k_{1},k_{2} \in \mathbb{N}$.
So, from Lemma \ref{gcddistance1;2} Part (5) we have that
$ \Delta_{m} = \Delta_{n} =d$. Since  $\Ft{0} =0$ and $\Ft{1} =1$, by Proposition \ref{modulo:dx} we have
\[
	\Ft{2k_{1}}  \equiv k_{1}g^{k_{1}-1} \modd{d} {d^2},
\]
\[
	\Ft{2k_{2}}  \equiv k_{2}g^{k_{2}-1} \modd{d} {d^2}.
\]
This and $\gcd(d ,g )=1$ imply that
\[\gcd(\Ft{m}\Ft{n-1},\Ft{m+1}\Ft{n},\Ft{m+2}\Ft{n-2} )=\gcd(k_{1}g^{k_{1}-1} d ,k_{2}g^{k_{2}-1} d ,d^2 )=d \gcd(d ,k_{1},k_{2}).\]
Similarly we have that
$\gcd(\Ft{m+1}\Ft{n-2},\Ft{m}\Ft{n} , \Ft{m+2}\Ft{n-1})	=d \gcd(d ,k_{1}-1,k_{2}+1)$.

Let $\beta=\left(\gcd(d ,k_{1}-1,k_{2}+1)\right)/\left(\gcd(d ,k_{1},k_{2})\right)$. Therefore,
\[\gcd(\Ft{m+1}\Ft{n-2},\Ft{m}\Ft{n} , \Ft{m+2}\Ft{n-1})=\beta\gcd(\Ft{m}\Ft{n-1},\Ft{m+1}\Ft{n},\Ft{m+2}\Ft{n-2} ).\]

We now prove Part (2). Factoring, we have that
\[
\gcd(\Ft{m+1}\Ft{n-2} ,\Ft{m+1}\Ft{n} )\cdot \gcd(\Ft{m} \Ft{n-1} ,\Ft{m+2}\Ft{n-1} )
\]
is equal to
\[ \Ft{m+1} \Ft{n-1} \cdot  \gcd(\Ft{n-2} ,\Ft{n} )\cdot \gcd(\Ft{m}  ,\Ft{m+2} ).\]
The conclusion follows using Lemma \ref{gcddistance1;2} Part (5).
\end{proof}

\begin{proof}[Proof of Theorem \ref{gcdstarofdavid:Part:3}.] In Lemma \ref{lem:Bts} if we take $G_{n} =\Lt{n} $, we have

\begin{equation}\label{Equat:Proof:Thm3:1}
\gcd(\Lt{m} \Lt{n-1},\Lt{m+1} \Lt{n}, \Lt{m+2} \Lt{n-2})=\gcd(\Lt{n} ,\Lt{m}, \Delta_m \Delta_n)
\end{equation}
and
\[
\gcd(\Lt{m+1} \Lt{n-2},\Lt{m} \Lt{n} ,\Lt{m+2} \Lt{n-1} )=\gcd(\Lt{n-2} ,\Lt{m+2} , \Delta_n  \Delta_m),
\]
where $\Delta_t=\gcd(\Lt{t}, \Lt{t-2})$.
If $m$ and $n$ are not both odd, then the  proof follows in a similar way as in the proof of Theorem \ref{gcdstarofdavid:Part:2}.

Suppose that both $m$ and $n$ are odd, that is  $n=2k_{1}+1$ and  $m=2k_{2}+1$ where $k_{1}, k_{2}$ are non-negative integers. Therefore,  by
Lemma \ref{gcddistance1;2} Part (4) we know that $ \Delta_{m} = \Delta_{n} =\Lt{1}$. Since $\Lt{1} |d $, by Proposition \ref{modulo:dx}
we have
\[
	\Lt{n} \equiv n g^{k_{1}} \modd{\Lt{1}} {\Lt{1}^2}
\]
and
\[
	\Lt{m} \equiv m g^{k_{2}} \modd{\Lt{1}} {\Lt{1}^2}.
\]
This and \eqref{Equat:Proof:Thm3:1} imply that
\[\gcd(\Lt{m} \Lt{n-1},\Lt{m+1} \Lt{n},\Lt{m+2} \Lt{n-2})=\gcd(n g^{k_{1}} \Lt{1} ,m g^{k_{2}} \Lt{1} , {(\Lt{1} )^2}).\]
This and $\gcd(d ,g )=1$ imply that
$ \gcd(\Lt{m} \Lt{n-1},\Lt{m+1} \Lt{n},\Lt{m+2} \Lt{n-2})=\Lt{1} \gcd(n, m, \Lt{1})$.

Similarly we can prove that
\[\gcd(\Lt{m+1} \Lt{n-2},\Lt{m} \Lt{n} ,\Lt{m+2} \Lt{n-1} )=\Lt{1} \gcd(\Lt{1} ,n-2,m+2).\]
Let $\beta^{\prime}=\left(\gcd(\Lt{1} ,n-2,m+2)\right)/\left(\gcd(\Lt{1} ,n,m)\right)$. Then,
\[\gcd(\Lt{m+1} \Lt{n-2},\Lt{m} \Lt{n} ,\Lt{m+2} \Lt{n-1} )= \beta^{\prime}\gcd(\Lt{m} \Lt{n-1},\Lt{m+1} \Lt{n},\Lt{m+2} \Lt{n-2}).\]

We now prove Part (2). Factoring, we have that
\[
\gcd(\Lt{m+1}\Lt{n-2} ,\Lt{m+1}\Lt{n} )\gcd(\Lt{m} \Lt{n-1} ,\Lt{m+2}\Lt{n-1} )
\]
is equal to
\[
\Lt{m+1} \Lt{n-1}  \gcd(\Lt{n-2} ,\Lt{n} )\gcd(\Lt{m}  ,\Lt{m+2} ).
\]
The conclusion follows using Lemma \ref{gcddistance1;2} Part (4).
\end{proof}

\subsection{Corollaries of the main theorem}

Theorems \ref{gcdstarofdavid:Part:2}, \ref{gcdstarofdavid:Part:3}, and \ref{gcdstarofdavid:Part1}
are also true for the star of David with a vertical configuration as depicted in Figure \ref{starofDavidF}
(with similar proofs). The following corollaries are a formalization of some results that  are in the proofs
of Theorems \ref{gcdstarofdavid:Part:2}  and \ref{gcdstarofdavid:Part:3}. For the following three corollaries
we suppose that the points are as given in Theorems \ref{gcdstarofdavid:Part:2} and \ref{gcdstarofdavid:Part:3}
and Figure \ref{starofDavidHosoya}.

\begin{figure} [!ht]
\begin{center}
\includegraphics[width=25mm]{stard_vertical.eps}
\end{center}
\caption{Vertical star of David. } \label{starofDavidF}
\end{figure}

\begin{corollary} \label{special:case:1}   Let $G_{t} $ be one of the following polynomials: Fibonacci, Lucas,
 Jacobsthal, Jacobsthal-Lucas,  Chebyshev first kind polynomials, Pell-Lucas, and both Morgan-Voyce polynomials, for every
 $t \in \mathbb{N}$. If $G_{m+1}G_{n-2}$, $G_{m}G_{n}$, and $G_{m+2}G_{n-1}$  are the points in a triangle of the
 star of David and  $G_{m}G_{n-1}$,  $G_{m+2}G_{n-2}$, and  $G_{m+1}G_{n}$  are the points in the second triangle of the star
 of David, then
	 $$\gcd(G_{m+1} G_{n-2} , G_{m} G_{n} , G_{m+2} G_{n-1} )=\gcd(G_{m} G_{n-1} , G_{m+2} G_{n-2} , G_{m+1} G_{n} ).$$ 	
\end{corollary}

\begin{corollary}\label{special:case:2}  Suppose that $\Ft{m+1}\Ft{n-2}$, $\Ft{m}\Ft{n}$, and $\Ft{m+2}\Ft{n-1}$
are the points in a triangle of the star of David and $\Ft{m}\Ft{n-1}$,  $\Ft{m+2}\Ft{n-2}$, and  $\Ft{m+1}\Ft{n}$
are the points in the second triangle of the star of David. If  $n=2k_{1}$ and $m=2k_{2}$ where $k_{1}, k_{2}\in \mathbb{N}$,
then the these hold

	\begin{enumerate}[(1)]
		\item if $n\ge 0$ and $\Ft{n}$  is a Pell polynomial or a Chebyshev polynomial of the second kind with
$k_{1}k_{2}\not\equiv \modd{0} {4} $ and $k_{1}\not\equiv \modd{k_{2}} {2} $, then
	$$\gcd(\Ft{m+1} \Ft{n-2} , \Ft{m} \Ft{n} , \Ft{m+2} \Ft{n-1} )=\gcd(\Ft{m} \Ft{n-1} , \Ft{m+2} \Ft{n-2} , \Ft{m+1} \Ft{n} ).$$
				
		\item If  $n\ge 0$ and $\Ft{n}$ is a Fermat polynomial with
        $k_{1}k_{2}\not\equiv \modd{0} {9} $ and $k_{1}\not\equiv 
	\modd{2 k_{2}} {3} $, then

	$$\gcd(\Ft{m+1} \Ft{n-2} , \Ft{m} \Ft{n} , \Ft{m+2} \Ft{n-1})=\gcd(\Ft{m} \Ft{n-1} , \Ft{m+2} \Ft{n-2} , \Ft{m+1} \Ft{n}).$$ 	
		
\end{enumerate}
\end{corollary}


\begin{corollary}\label{special:case:3}  Suppose that  $\Lt{m+1}\Lt{n-2}$, $\Lt{m}\Lt{n}$, and $\Lt{m+2}\Lt{n-1}$ are
the points in a triangle of the star of David and
 $\Lt{m}\Lt{n-1}$,  $\Lt{m+2}\Lt{n-2}$, and  $\Lt{m+1}\Lt{n}$ are the points in the second triangle of the star of David.
If $m,n \ge 0$,  $\Lt{t}$ is a Fermat-Lucas polynomial for $t\ge 0$, and  $\Lt{m}\Lt{n} \not = \Lt{0}\Lt{0}$,  then
	$$\gcd(\Lt{m+1} \Lt{n-2} , \Lt{m} \Lt{n} , \Lt{m+2} \Lt{n-1})=\gcd(\Lt{m}\Lt{n-1} , \Lt{m+2}\Lt{n-2}, \Lt{m+1} \Lt{n}).$$
 \end{corollary}


\section{The geometry of some identities}

The aim of this section is to give geometric interpretations of
some polynomial identities that are known for the Fibonacci numbers.  The novelty of this section is that we extend
some well-known numerical identities to $\{ \Ft{k} \}$ and to $\{ \Lt{k} \}$ sequences and provide geometric proofs for
these identities instead of the classical mathematical induction proofs.

Hosoya-type triangles (polynomial and numeric) are good tools to discover, prove, or represent theorems geometrically.
Some properties that have been found and proved algebraically are easy to understand when interpreted geometrically
using  these triangles.

\subsection{Identities in the Hosoya polynomial triangle}

\begin{lemma}\label{Propiedad:de:las:paralelas}
If $i$, $j$, $k$, and $r$ are nonnegative integers with $k+j\le r$, then in the Hosoya polynomial triangle this  holds
\[
	H(r+2i,k+j+i)-H(r+2i,k+i)=(-1)^ig (H(r,k+j)-H(r,k)).
\]
\end{lemma}

The proof of the Lemma \ref{Propiedad:de:las:paralelas} follows using induction
and the rectangle property which states that $H(n,m)=d H(n-1,m)+g H(n-2,m)$ (see Figure \ref{rectangle}).

\begin{figure} [!ht]
\begin{center}
\includegraphics[width=125mm]{Lemma9_Rectangle_property.eps}
\end{center}
\caption{Property of Rectangle.} \label{rectangle}
\end{figure}

It is well known that the Catalan identity is a generalization of the Cassini identity.  Johnson \cite{JohnsonC},
gives another numerical generalization of the Cassini and Catalan identities,
called the Johnson identity. It states that for the Fibonacci number sequence $\{F_{n}\}$,
$$F_{a}F_{b}-F_{c}F_{d}=(-1)^{r}\left(F_{a-r}F_{b-r}-F_{c-r}F_{d-r}\right)$$
 where $a, b, c, d$, and $r$ are arbitrary integers with $a+b=c+d$.

The example in Figure \ref{Cassini_Catalan} gives a geometric representation of the numeric identities
(the same representation holds for polynomials). To represent the Cassini identity we take two
consecutive points in the Hosoya triangle along a horizontal line such that one point is located
in the central column of the triangle, see Figure \ref{Cassini_Catalan}. We then pick two other
arbitrary consecutive points $P_{1}$ and $P_{2}$ such that they form a vertical rectangle along with the
first pair of points. The subtraction of the horizontal points $P_{1}$ and $P_{2}$ gives
$\pm 1$. Since the entries of the triangle are products of Fibonacci numbers, we obtain the Cassini identity.

The second example in Figure \ref{Cassini_Catalan} represents the Catalan identity. In this case we take any
two horizontal points $Q_{1}$ and $Q_{2}$ where $Q_{1}$ is located (arbitrarily) in the central column of the
triangle. We then pick other two arbitrary points $P_{1}$ and $P_{2}$ which form a rectangle with
$Q_{1}$ and $Q_{2}$. The subtraction of the horizontal points $P_{1}$ and $P_{2}$ gives $\pm (Q_{1}- Q_{2})$.
Since the entries of the triangle are products of Fibonacci numbers, we obtain the Catalan identity. Note that if we
eliminate the condition that $Q_{1}$ must be in the central column, we obtain the Johnson identity.

\begin{figure} [!ht]
\begin{center}
\includegraphics[width=150mm]{Corollary11_Rectangle_property.eps}
\end{center}
\caption{Cassini and Catalan identities in the Hosoya Triangle.} \label{Cassini_Catalan}
\end{figure}

\begin{theorem}\label{johnson}
Let $a, b, c, d$ and $t$ be nonnegative integers with $\min\{a,b,c,d\}-t$ non-negative.
Suppose that $G_k$ is either $\Ft{k}$ or $\Lt{k}$ for all $k\ge 0$. If $a+b=c+d$, then
\[
	\begin{vmatrix}
	G_{a}  &amp; G_{c}  \\
	G_{d}  &amp; G_{b}  \end{vmatrix}
	= (-1)^t g^t
	\begin{vmatrix} G_{a-t}  &amp; G_{c-t}  \\
	G_{d-t}  &amp; G_{b-t}  \end{vmatrix}.
	\]
\end{theorem}
\begin{proof}
Let $i$, $j$, $k$, and $r$ be nonnegative integers such that $a=k+j+i$, $b=r+i-k-j$, $c=k+i$, $d=r+i-k$, and $t=i$.
Therefore, by Lemma \ref{Propiedad:de:las:paralelas} and  Proposition \ref{lemma0} the equality holds.
\end{proof}

Theorem \ref{johnson} is a generalization of Johnson identity \cite{JohnsonC} and Falc\'on and Plaza identity \cite{Falcon}.
As a consequence of Theorem \ref{johnson} we state Corollary \ref{catalan:casini} --- this generalizes the Catalan
identity to $\Ft{k}$ and $\Lt{k}$. If  in Corollary \ref{catalan:casini} we take $r=1$, then we obtain a generalization of
Cassini identity.

\begin{corollary} [Catalan identity]\label{catalan:casini}
	Suppose that $m, r$ are non-negative integers.
	If $G_k$ is either $\Ft{k}$ or $\Lt{k}$ for all $k\ge 0$, then  	
		\[
		\left|\begin{array}{ll}
		G_{m}  &amp; G_{m+r}  \\
		G_{m-r}  &amp; G_{m}
		\end{array}\right|
		=(-1)^{m-r} g^{m-r}
		\left|\begin{array}{ll} G_{r}  &amp; G_{2r}  \\
		G_{0}  &amp; G_{r}
		\end{array}\right|.
		\]	
		
		
\end{corollary}

\begin{proof}  The proof is straightforward when the appropriate values of $m$ and $r$ are substituted in
Theorem \ref{johnson} (see Figure \ref{Cassini_Catalan}). If we evaluate both determinants in Theorem \ref{johnson}
we obtain four summands that are four points in the Hosoya polynomial triangle. Note that these four points are the
vertices of a rectangle in the Hosoya triangle.
\end{proof}

 \begin{figure} [!ht]
\begin{center}
\includegraphics[width=100mm]{Theorem12_Application_Rectangle_property.eps}
\end{center}
\caption{Geometric interpretation of Theorem \ref{sums:property}. } \label{rectangleTheorem12}
\end{figure}

We observe that if we have a Hosoya triangle where the entries are products  of two polynomial of  $\{ \Ft{k} \}$,
then we can draw rectangles with two vertices in the central line (the perpendicular bisector) of the triangle and
a third vertex on the edge of the triangle (see Figure \ref{rectangleTheorem12}). For a fixed $i \in \mathbb{N}$ let $R_i$
be a rectangle  with the extra condition that the upper vertex points are multiplied by $g $, then
Lemma \ref{Propiedad:de:las:paralelas} guarantees that the sum of the two top vertices of $R_i$  is equal to
the sum of the remaining vertices of $R_i$. Since the points in the edge of this triangle are equal to zero,
one of the vertices of $R_i$ is equal to zero. The other vertex in the same vertical line is a polynomial $\Ft{i}$
multiplied by one. This geometry gives rise to Theorem \ref{sums:property}.

For the next result we introduce the following function. We recall that $g$ is as defined in \eqref{Fibonacci;general:FT}.
 \[
 I(n)=
 \begin{cases}
         g , &amp; \mbox{if $n$ is even;} \\
         1, &amp; \mbox{if $n$ is odd.}
\end{cases}
 \]

\begin{theorem}\label{sums:property} If $n$ and $k$ are positive integers, then
	$$
	\sum_{j=2}^{2n+1} I(j)\Ft{j}^{2}  =\sum_{j=1}^n \Ft{4j+1}
	$$
	and
	$$
	\sum_{j=2}^{2n+1} (-1)^{j+1}I^2(j)\Ft{2j}^{2}  =  d \sum_{j=1}^{n}\Ft{8j+2}  .
	$$
\end{theorem}

\begin{proof} First of all we recall that $\Ft{1} =1$. We prove the first identity.
	\begin{eqnarray*}
		\sum_{j=2}^{2n+1} I(j)\Ft{j}^{2}  &amp;=&amp;\sum_{j=1}^{n} (\Ft{2j+1}^{2} + g \Ft{2j}^{2}  )
		=\sum_{j=1}^n (\Ft{4j+1} \Ft{1} +\Ft{0} \Ft{4j})\\
		&amp;=&amp;\sum_{j=1}^n \Ft{4j+1} .
	\end{eqnarray*}
	
We now prove the second identity.  Let  $S:=\sum_{j=2}^{2n+1} (-1)^{j+1}I^2(j)\Ft{2j}^{2}$.
Lemma \ref{Propiedad:de:las:paralelas}  implies that
	\begin{eqnarray*}
		S	&amp;=&amp;\sum_{j=1}^{n} (\Ft{4j+2}^{2} - g^2 \Ft{4j}^{2}  )\\
		&amp;=&amp;\sum_{j=1}^{n} \Big(\Big(\Ft{4j+2}^{2} +g \Ft{4j+1}^{2} \Big)-g \Big(\Ft{4j+1}^{2}  +g \Ft{4j}^{2}  \Big)\Big).\\
	\end{eqnarray*}	
Since $\Ft{1} =1$, we have
\begin{eqnarray*}
		S	&amp;=&amp;\sum_{j=1}^{n} \left(\left(\Ft{8j+3} +g \Ft{8j+1} \Ft{0} \Big)-g \Big(\Ft{8j+1} +g \Ft{8j-1} \Ft{0} \right)\right)\\
		&amp;=&amp; \Ft{1} \sum_{j=1}^{n} \Big(\Ft{8j+3} -g \Ft{8j+1} \Big)=d \sum_{j=1}^{n}\Ft{8j+2}.
	\end{eqnarray*}
	This completes the proof.
\end{proof}

Corollary \ref{Corollay:sums:property} provides a closed formula for special cases of Theorem \ref{sums:property}.
 We use Figure \ref{rectangleCorollary14} to give a geometric interpretation of Corollary  \ref{Corollay:sums:property}.
 For brevity  we only give an algebraic proof of Part (1),  the algebraic proof of Part (2) is similar, therefore it is
 omitted, and instead we provide a geometric proof of Part (2). This gives us the geometric behavior of a zigzag pattern
 of points.  Thus, Corollary   \ref{Corollay:sums:property} Part (2) states that the sum of all points that are in the
 intersection of a finite zigzag configuration and the central line of the triangle is the last point of the zigzag
 configuration (see Figure \ref{rectangleCorollary14}).

\begin{corollary}\label{Corollay:sums:property} Suppose that $g$ is as defined in \eqref{Fibonacci;general:FT}. Then these hold
	
	\begin{enumerate}[(1)]
		\item 	\[\sum_{j=1}^{n}g^{2n-j}\Ft{4j-3} =\frac{\Ft{2n-1} \Ft{2n}}{d }.\]
	
			\item  If  (in particular)  the sequence $\{ \Ft{k} \}$ satisfies that $g=1$, then
		\[	\sum_{j=1}^{2n-1} \Ft{j}^{2} =\frac{\Ft{2n-1} \Ft{2n}}{d }.\]
	\end{enumerate}	
\end{corollary}

\begin{figure} [!ht]
	\begin{center}
		\includegraphics[width=120mm]{Corollary14_Application_Rectangle_Pro.eps}
	\end{center}
	\caption{Geometric interpretation of Corollary \ref{Corollay:sums:property}. } \label{rectangleCorollary14}
\end{figure}

\begin{proof}
Since $H(2n,n)=  \Ft{n}^2 $, we have that   $g^n H(1,1)+\sum_{j=1}^{n} d g^{n-j}  \Ft{j}^2 $ is equal to
\begin{eqnarray*}
	\sum_{j=1}^{n} d g^{n-j}H(2j,j)
	&amp;=&amp;g^{n-1} (g H(1,1)+d H(2,1))+\sum_{j=2}^{n} d g^{n-j}  \Ft{j}^2 \\
	&amp;=&amp;g^{n-1} H(3,1)+d g^{n-2}H(4,2)+\sum_{j=3}^{n} d g^{n-j}  \Ft{j}^2 \\
	&amp;=&amp;g^{n-2} H(5,3)+\sum_{j=3}^{n} d g^{n-j}  \Ft{j}^2 \\
	&amp;=&amp;g^{n-2}   \Ft{3}   \Ft{2} +\sum_{j=3}^{n} d g^{n-j}  \Ft{j}^2 .
\end{eqnarray*}
Similarly, we find that
\begin{equation}\label{general}
\sum_{j=1}^{n} d g^{n-j}  \Ft{j}^2 =H(2n+1,n+1)=  \Ft{n+1}   \Ft{n} -g^n   \Ft{1}   \Ft{0} .
\end{equation}
Note that
\begin{eqnarray*}
	\sum_{j=2}^{2n+1} g^{2n+1-j} \Ft{j}^{2}  &amp;=&amp;\sum_{j=1}^{n} g^{2n-j} (\Ft{2j+1}^{2} + g \Ft{2j}^{2}  )\\
	&amp;=&amp;\sum_{j=1}^n g^{2n-j} (\Ft{4j+1} \Ft{1} +\Ft{0} \Ft{4j})\\
	&amp;=&amp;\sum_{j=1}^n g^{2n-j} \Ft{4j+1}.
\end{eqnarray*}
This, Equation \eqref{general}, and $  \Ft{0} =0$ complete the proof of Part (1).

Proof of Part (2). From the hypothesis of  Part (2), $g=1$,  we see that the sequence $\{\Ft{n}\}$
defined in \eqref{Fibonacci;general:FT} satisfies that  $g=1$ and  that $H(0,k)=H(k,0)=0$ for every $k$. This and the definition
of the Hosoya polynomial sequence (page~\pageref{HosoyaSection}), imply that
$$H(r,k)= d  H(r-1,k)+ H(r-2,k)    \text{ and  } H(r,k)= d  H(r-1,k-1)+H(r-2,k-2).$$
Therefore the points depicted in Figure \ref{rectangleCorollary14} have the properties described in Table \ref{PointZigZag}.
\begin{table}[!ht]
	\begin{center}
		\begin{tabular}{llll}
			$p_{0}=0$,                    		&amp;$p_{2}=d p_{1}+p_{0}$,        		&amp; $p_{4}=d p_{3}+p_{2}$ &amp; $p_{6}=d p_{5}+p_{4}$ \\
			$p_{8}=d p_{7}+p_{6}$,\quad \quad &amp; $p_{10}=d p_{9}+p_{8}$, \quad\quad&amp;  \hspace{1cm}\dots  &amp; $p_{4n}=d p_{4n-1}+p_{4n-2}$.
		\end{tabular}
	\end{center}
	\caption{Properties of points in the Zigzag Figure \ref{rectangleCorollary14}.} \label{PointZigZag}
\end{table}

Since $g=1$, we have that $I(j)=1$ for all $j$. Therefore, $\sum_{j=1}^{2n+1} I(j)\Ft{j}^{2}=\sum_{j=1}^{2n+1} \Ft{j}^{2} $
is actually the sum of all points that are in the intersection of the  zigzag diagram with central line of the triangle
(see Figure \ref{rectangleCorollary14}). Thus,
$$d \sum_{j=1}^{2n+1} \Ft{j}^{2}  =p_{0}+d p_{1}+d p_{3}+d p_{5}+d p_{7}+\dots+d p_{4n-1}.$$
The sum of the first two terms in the right side is equal to
the third point of the  zigzag diagram (see Table \ref{PointZigZag} and Figure \ref{rectangleCorollary14}). Therefore, substituting them with $p_{2}$ we have
$$d \sum_{j=1}^{2n+1} \Ft{j}^{2}  =p_{2}+d p_{3}+d p_{5}+d p_{7}+\dots+d p_{4n-1}.$$
 Now the sum of the first two terms of the right side of the previous equation is equal to
the fifth point, $p_{4}$, of the  zigzag diagram (see Table \ref{PointZigZag} and Figure \ref{rectangleCorollary14}).
Therefore, substituting them with $p_{4}$ we have
 $$d \sum_{j=1}^{2n+1} \Ft{j}^{2}  =p_{4}+d p_{5}+d p_{7}+\dots+d p_{4n-1}.$$
Similarly, we substitute $p_{4}+d p_{5}$ with the seventh point of the  zigzag diagram. Thus,
 $$d \sum_{j=1}^{2n+1} \Ft{j}^{2}  =p_{6}+d p_{7}+\dots+d p_{4n-1}.$$
Continuing this process,  systematically  substituting the terms, we obtain
$$d \sum_{j=1}^{2n+1} \Ft{j}^{2}  =p_{4n}=\Ft{2n-1}\Ft{2n}.$$ This completes the geometric proof of Part (2).
\end{proof}

\subsection{Integration in the Hosoya triangle}
We now discuss some examples on how the geometry of the triangle can be used to represent identities. The examples given
in the following discussion are only for the case in which the Hosoya triangle (denoted by $H_{F} $) has
products of Fibonacci polynomials as entries. With this triangle in mind  we introduce a notation that will
be used in following examples.  We define an $n$-initial triangle as the finite triangular arrangement
formed by the first $n$-rows of $H_{F} $ with non-zero entries. Note that the initial triangle
is the equilateral sub-triangle of the Hosoya triangle as in Table \ref{tabla_equivalent} on page~\pageref{tabla_equivalent}
without the entries containing the factor $G_{0}$. For instance, Table \ref{tabla2} on page~\pageref{tabla2} represents the
$5$-initial triangle of $H_{F}$.

If $F_{n}^{\prime}(x)$ represents the derivative of the  Fibonacci polynomial $F_{n}(x)$, then
$F_{n}^{\prime}(x) =\sum_{k=1}^{n-1}F_{k}(x)F_{n-k}(x)$ (see \cite{Falcon,koshy}). The geometric  representation of this property in  an
$n$-initial triangle is as follows: the derivative of the first entry of the last row of a given $n$-initial triangle
 is equal to the sum of all points of the penultimate row of this triangle (see Table \ref{tabla2} on page~\pageref{tabla2}).
 We have observed that this property implies that the integral of all points of the first $n-1$ rows of a given $n$-initial triangle
  is equal to the sum of all  points of one edge of this triangle, where the constant of integration is $\lceil{n/2}\rceil$. This result is stated formally in Proposition \ref{integral}. Similar results can be obtained using Table \ref{integral:Fibonacci:typeP}.

\begin{proposition}\label{integral} Let $n$ be a positive number, then

\begin{enumerate}[(1)]
\item \begin{eqnarray*}
H(n,1)&amp;=&amp;\sum_{k=1}^{n-1} \int {H(n-1,k)}.
\end{eqnarray*}
Equivalently,
 \begin{eqnarray*}
F_{n}(x)&amp;=&amp;\sum_{k=1}^{n-1} \int {F_{k}(x)F_{n-k}(x)},
\end{eqnarray*}
where the constant of integration is $C=1$ if $n$ is odd and zero otherwise.

\item
\begin{eqnarray*}
	H(n+1,1)+H(n,1)-1		&amp;=&amp;x \sum_{r=1}^{n} \sum_{k=1}^{r-1} \int {H(r-1,k)}.
	\end{eqnarray*}
Equivalently,
 \begin{eqnarray*}
	F_{n+1}(x)+F_{n}(x)-1	&amp;=&amp;x\sum_{r=1}^{n} \sum_{k=1}^{r-1} \int {F_{k}(x)F_{r-k}(x)},
\end{eqnarray*}
where the constant of integration is $C=\lceil{n/2}\rceil$.

\end{enumerate}
\end{proposition}

\begin{proof}
The proof of Part (1) is straightforward using the geometric interpretation of $F_{n}^{\prime}(x)$.

We prove Part (2). From Part (1) and from the geometry of  the $(n-1)$-initial triangle we have
$\sum_{r=1}^{n} H(k,1)=\sum_{r=1}^{n-1} \sum_{k=1}^{r-1} \int {H(r-1,k)}$.
From Koshy \cite[Theorem 37.1]{koshy} we know that
$F_{n+1}(x)+F_{n}(x)-1=x\sum_{i=1}^{n} F_{i}(x)$.
This and the fact that $H(t,1)=F_{t}(x)$ for all $t\ge 1$ completes the proof.
\end{proof}

\begin{table}[!ht]
\begin{center}
\begin{tabular}{|l|}
\hline
Derivative 		\\ \hline
$F_n^{\prime}(x)=\sum_{k=1}^{n-1}F_{k}(x)F_{n-k}(x)$ 	\\ [5pt]
$P_n^{\prime}(x)=2 \sum_{k=1}^{n-1}P_{k}(x)P_{n-k}(x)$	\\ [5pt]
$\Phi_n^{\prime}(x)=3\sum_{k=1}^{n-1}\Phi_{k}(x)\Phi_{n-k}(x)$	\\ [5pt]
$U_n^{\prime}(x)=2 \sum_{k=1}^{n-1}P_{k}(x)P_{n-k}(x)$	\\ [5pt]
$B_n^{\prime}(x)=\sum_{k=1}^{n-1}B_{k}(x)B_{n-k}(x)$\\
\hline
\end{tabular}
\end{center}
\caption{Derivatives of Fibonacci-type polynomials.} \label{integral:Fibonacci:typeP}
\end{table}

\section{Appendix. Numerical types of Hosoya triangle}

In this section we study some connections of the Hosoya polynomial triangles with some numeric sequences
that may be found in \cite{sloane}. We show that when we evaluate the entries of a Hosoya polynomial triangle
at $x=1$ they give a triangle that is in \url{http://oeis.org/}. The first Hosoya triangle is the classic Hosoya
triangle formerly  called the Fibonacci triangle.

We now introduce some notation that is used in Table \ref{Tableahosoyatriangles}. Recall that $H_{F} $
is the Hosoya triangle with products of Fibonacci polynomials as entries. Similarly
we define the notation for the Hosoya polynomial triangle of the other types
--- Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell polynomials,
Fermat polynomials, Jacobsthal polynomials. The star of David property holds obviously for all these
numeric triangles.

\begin{table}[!ht]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
Triangle type 		&amp; Notation 		&amp;  Entries 					&amp; Sloane 			  	\\ \hline
Fibonacci 	    		&amp; $H_F(1)$		&amp; $F_k(1)F_{r-k}(1)$      		&amp;  \seqnum{A058071} 	\\
Lucas 	    		&amp;$H_D(1)$		&amp; $D_k(1)D_{r-k}(1)$      		&amp;\seqnum{A284115} 	\\
Pell		    		&amp; $H_P(1)$		&amp; $P_k(1)P_{r-k}(1)$      		&amp;  \seqnum{A284127} 	\\
Pell-Lucas    		&amp;$H_Q(1)$		&amp; $Q_k(1)Q_{r-k}(1)$      		&amp;  \seqnum{A284126}	\\
Fermat 	    		&amp;$H_{\Phi}(1)$	&amp; $\Phi_k(1)\Phi_{r-k}(1)$		&amp; \seqnum{A143088} 	\\
Fermat-Lucas 		&amp;$H_{\vartheta}(1)$	&amp; $\vartheta_k(1)\vartheta_{r-k}(1)$ &amp; \seqnum{A284128} \\
Jacobsthal   		&amp;$H_J(1)$		&amp; $J_k(1)J_{r-k}(1)$      		&amp;\seqnum{A284130} 	\\
Jacobsthal-Lucas 	&amp; $H_j(1)$		&amp; $j_k(1)j_{r-k}(1)$      		&amp; \seqnum{A284129} 	\\
Morgan-Voyce		&amp; $H_B(1)$		&amp; $B_k(1)B_{r-k}(1)$      		&amp;\seqnum{A284131}		\\
Morgan-Voyce 		&amp;$H_C(1)$		&amp; $C_k(1)C_{r-k}(1)$      		&amp;\seqnum{A141678}		\\
\hline
\end{tabular}
\end{center}
\caption{Numerical Hosoya triangles present in Sloane \cite{sloane}.} \label{Tableahosoyatriangles}
\end{table}

We also observe some curious numerical patterns when we compute the gcd of the coefficients of polynomials
discussed in this paper. In particular, the gcd of the coefficients of $\Phi_{n}(x)$, the $n$th Fermat
polynomial, is $3^{a_{n}}$ where $a_{n}$ is the $n$th element of \seqnum{A168570}. The gcd of the coefficients
of $\vartheta_{n}(x)$, the $n$th Fermat-Lucas polynomial, is $3^{a_{n}}$ where $a_{n}$ is the $n$th element of
\seqnum{A284413}. We also found that the gcd of the coefficients of the $P_{2n}(x)$, the $2n$th
Pell polynomial, is $2^{a_{n}}$ where $a_{n}$ is the $n$th element of \seqnum{A001511}.
Finally, the gcd of the coefficients of the $U_{n}(x)$, the $n$th Chebyshev polynomial of second kind,
is $2^{a_{n}}$ where $a_{n}$ is the $n$th element of \seqnum{A007814}.

\section{Acknowledgment}

The first and last authors were partially supported by The Citadel Foundation. The authors are grateful to an
anonymous referee for the extensive comments and suggestions that helped to improve the presentation of this paper.

\begin{thebibliography}{99}

\bibitem{Falcon} S.~Falc\'on and A.~Plaza, On $k$-Fibonacci sequences and polynomials and their derivatives,
\emph{Chaos Solitons Fractals} \textbf{39} (2009), 1005--1019.

\bibitem{florezHiguitaMukCharact} R.~Fl\'{o}rez, R.~Higuita, and A.~Mukherjee,
Characterization of the strong divisibility property for generalized Fibonacci polynomials,
\emph{Integers} \textbf{18} (2018),  Paper No. A14.

\bibitem{florezHiguitaMuk} R.~Fl\'{o}rez, R.~Higuita, and A.~Mukherjee,
Alternating sums in the Hosoya polynomial triangle, 
\emph{J. Integer Seq.} \textbf{17} (2014), Article 14.9.5.

\bibitem{florezHiguitaJunes} R.~Fl\'{o}rez, R.~Higuita, and L.~Junes,
9-modularity and GCD properties of generalized Fibonacci numbers,
\emph{Integers} \textbf{14} (2014),  Paper No. A55.

\bibitem{florezHiguitaJunesGCD} R.~Fl\'{o}rez, R.~Higuita, and L.~Junes,
GCD property of the generalized star of David in the generalized Hosoya triangle,
\emph{J. Integer Seq.} \textbf{17} (2014),  Article 14.3.6.

\bibitem{florezjunes} R.~Fl\'{o}rez and L.~Junes,
GCD properties in Hosoya's triangle, \emph{Fibonacci Quart.} \textbf{50} (2012), 163--174.

\bibitem{hoggattHillman} A.~P.~Hillman and V.~E.~Hoggatt, Jr.
A proof of Gould's Pascal hexagon conjecture, \emph{Fibonacci Quart.} \textbf{6} (1972), 565--568, 598.

\bibitem{hoggatt_star_david} V.~E.~Hoggatt, Jr., and W.~Hansell,
 The hidden hexagon squares,
 \emph{Fibonacci Quart.} \textbf{9} (1971), 120, 133.

\bibitem{Pell} A.~F.~Horadam and J.~M.~Mahon, Pell and Pell-Lucas polynomials,
\emph{Fibonacci Quart.} \textbf{23} (1985), 7--20.

\bibitem{Fermat} A.~F.~Horadam, Chebyshev and Fermat polynomials for diagonal
functions, \emph{Fibonacci Quart.}  \textbf{17}  (1979), 328--333.

\bibitem{hosoya} H.~Hosoya, Fibonacci triangle, \emph{Fibonacci Quart.}  \textbf{14} (1976), 173--178.

\bibitem{JohnsonC} R.~C.~Johnson, Fibonacci numbers and matrices, preprint
available at
\url{http://maths.dur.ac.uk/~dma0rcj/PED/fib.pdf}.

\bibitem{koshygibonomial} T.~Koshy, Gibonomial coefficients with interesting byproducts,
\emph{Fibonacci Quart.} \textbf{53} (2015), 340--348.

\bibitem{koshy} T.~Koshy,
\emph{Fibonacci and Lucas Numbers with Applications},
Wiley, 2001.

\bibitem{koshy1} T.~Koshy, \emph{Fibonacci and Lucas Numbers with
Applications},  Vol.~2, preprint.

\bibitem{koshytriangular} T.~Koshy, \emph{Triangular Arrays with Applications},
Oxford University Press, 2011.

\bibitem{Korntved} E.~Korntved,
Extensions to the GCD star of David theorem, \emph{Fibonacci Quart.} \textbf{32} (1994), 160--166.

\bibitem{Long} C.~Long, W.~Schulz, and S.~Ando,
An extension of the GCD Star of David theorem, \emph{Fibonacci Quart.} \textbf{45} (2007), 194--201.

\bibitem{Sagan} B.~Sagan and C.~D.~Savage,
Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences,
\emph{Integers} \textbf{10} (2010), 697--703.

\bibitem{sloane} N.~J.~A.~Sloane, The On-Line Encyclopedia of Integer Sequences, \url{http://oeis.org/}.

\bibitem{sun} Y.~Sun, The star of David rule,
\emph{Linear Algebra Appl.} \textbf{429} (2008), 8--9, 1954--1961.

\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B39; Secondary 11B83.

\noindent \emph{Keywords: }
Hosoya triangle, Gibonomial triangle, Fibonacci polynomial, Chebyshev
polynomial, Morgan-Voyce polynomial, Lucas polynomial, Pell polynomial,
Fermat polynomial.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A001511},
\seqnum{A007814},
\seqnum{A058071},
\seqnum{A141678},
\seqnum{A143088},
\seqnum{A168570},
\seqnum{A284115},
\seqnum{A284126},
\seqnum{A284127},
\seqnum{A284128},
\seqnum{A284129},
\seqnum{A284130},
\seqnum{A284131}, and
\seqnum{A284413}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received May 31 2017;
revised versions received  March 5 2018; March 20 2018; April 15 2018.
Published in {\it Journal of Integer Sequences}, May 8 2018.

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\noindent
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