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\begin{center}
\vskip 1cm{\LARGE\bf 
A Formula for the Generating Functions of
\vskip .05in
Powers of Horadam's Sequence with Two
\vskip .13in
Additional Parameters
}
\vskip 1cm
\large
Emrah K\i l\i \c{c} \\
TOBB University of Economics and Technology\\
Mathematics Department\\
06560 Ankara\\
Turkey\\
\href{mailto:ekilic@etu.edu.tr}{\tt ekilic@etu.edu.tr} \\
\ \\
Y\"{u}cel T\"{u}rker Uluta\c{s} and Ne\c{s}e \"{O}m\"{u}r \\
Kocaeli University\\
Mathematics Department \\
41380 \.{I}zmit Kocaeli  \\
Turkey \\ 
\href{mailto:turkery@kocaeli.edu.tr}{\tt turkery@kocaeli.edu.tr} \\
\href{mailto:neseomur@kocaeli.edu.tr}{\tt neseomur@kocaeli.edu.tr} \\
\end{center}

\vskip .2 in

\begin{abstract}
     In this note, we give a generalization of a formula for the
generating function of powers of Horadam's sequence by adding two
parameters. Thus we obtain a generalization of a formula of Mansour.
\end{abstract}

\section{Introduction}

Horadam \cite{1,2} defined the second-order linear recurrence
sequence $\left\{ W_{n}\left( a,b;p,q\right) \right\} ,$ or briefly
$\left\{ W_{n}\right\} , $ as follows:
\begin{equation}
W_{n+1}=pW_{n}+qW_{n-1,}\text{ \ \ \ \ }W_{0}=a,W_{1}=b  
\end{equation}%
where $a,$ $b$ and $p,q$ are arbitrary real numbers for $n>0~$.
The Binet formula for the sequence $\left\{ W_{n}\right\} $ is%
\begin{equation*}
W_{n}=\dfrac{A\alpha ^{n}-B\beta ^{n}}{\alpha -\beta }\ ,
\end{equation*}%
where $A=b-a\beta $ and $B=b-a\alpha .$ When $a=0,$ $b=1,$ and, $a=2,$ $b=1,$ denote $W_{n}$ by $U_{n}$ and $%
V_{n},$ respectively. If we take $p=1,q=1,$ then $U_{n}=F_{n}$
($n$th Fibonacci number) and $V_{n}=L_{n}$ ($n$th Lucas number).

K\i l\i \c{c} and Stanica \cite{8} showed that for $r>0,n>0$, the
sequence $\left\{
W_{n}\right\} $ satisfies the following recursion%
\begin{equation*}
W_{r\left( n+2\right) }=V_{r}W_{r\left( n+1\right) }-\left(
-q\right) ^{r}W_{rn}.
\end{equation*}

 Riordan \cite{4} found the generating function for powers of
Fibonacci numbers. He proved that the generating function $S
_{k}\left( x\right) =\sum_{n\geq 0}F_{n}^{k}x^{n}$ satisfies the
recurrence relation
\begin{equation*}
\left( 1-a_{k}x+\left( -1\right) ^{k}x^{2}\right) S _{k}\left(
x\right) =1+kx\sum_{j=1}^{\left[ k/2\right] }\left( -1\right)
^{j}\frac{a_{kj}}{j}S _{k-2j}\left( \left( -1\right) ^{j}x\right) ,
\end{equation*}%
for $k\geq 1,$ where $a_{1}=1,~a_{2}=3,~a_{s}=a_{s-1}+a_{s-2}$ for
$s\geq 3,$ and $\left( 1-x-x^{2}\right) ^{-j}=\sum_{k\geq
0}a_{kj}x^{k-2j}.$ Horadam \cite{2} gave a recurrence relation for
$H_{k}\left( x\right) ~$(see also \cite{5}). Haukkanen \cite{6}\
studied linear combinations of Horadam's sequences and the
generating function of the ordinary product of two of Horadam's
sequences.

Mansour \cite{3} studied about the generating function for powers of
Horadam's sequence given by $H_{k}\left( x;a,b,p,q\right)
=H_{k}\left( x\right) =\sum_{n\geq 0}W_{n}^{k}x^{n}.$ Then he showed
that the generating function $H_{k}\left( x\right) $ can be
expressed the ratio of two $k$ by $k$ determinants as well as he
gave some applications for the generating function $H_{k}\left(
x\right) $.

In this study, we consider the generating function for powers of
Horadam's sequence defined by
\begin{equation*}
\Re _{k,t,r}\left( x;a,b,p,q\right) =\Re _{k,t,r}\left( x\right)
=\sum_{n\geq t}W_{rn}^{k}x^{n}.
\end{equation*}%
We shall derive a ratio to express the generating function $\Re
_{k,t,r}\left( x\right)$ by using the method of Mansour. Moreover,
we give applications of our results.

\section{The Main Result}
Firstly, we define two $k$ by $k$ matrices, in order to express the
$\Re _{k,t,r}\left( x\right) $ as a ratio of two determinants. Let
$\Delta _{k,r}=\left( \Delta _{k,r}\left( i,j\right) \right) _{1 \leq
i,j \leq k}=\Delta _{k,r}(p,q)$ be the $k\times k$ matrix have the
form

\begin{eqnarray*}
&&\Delta _{k,r}(p,q) \\
&=&\left[
\begin{array}{cccccc}
1-xv_{r}^{k}-x^{2}\left( -\left( -q\right) ^{r}\right) ^{k} &
-xv_{r}^{k-1}\left( -\left( -q\right) ^{r}\right) \binom{k}{1} & ...
&
-xv_{r}\left( -\left( -q\right) ^{r}\right) ^{k-1}\binom{k}{k-1} \\
-v_{r}^{k-1}x & 1-xv_{r}^{k-2}\left( -\left( -q\right) ^{r}\right) \binom{k-1%
}{1} & ...
 &
-x\left( -\left( -q\right) ^{r}\right) ^{k-1}\binom{k-1}{k-1} \\
-v_{r}^{k-2}x & -xv_{r}^{k-3}\left( -\left( -q\right) ^{r}\right) \binom{k-2%
}{1}&  ...  &0 & \\
... & ... & ... & ...  \\
-v_{r}^{2}x & -xv_{r}\left( -\left( -q\right) ^{r}\right)
\binom{2}{1}& ... &0 &
   \\
-v_{r}x & -x\left( -\left( -q\right) ^{r}\right) \binom{1}{1}&...
&1&
%
\end{array}%
\right]
\end{eqnarray*}%
and let $\delta _{k,t,r}=$ $\delta _{k,t,r}(a,b,p,q)$ be the
$k\times k$ matrix have the form


\begin{eqnarray*}
&&\delta _{k,t,r}(a,b,p,q) \\
&=&\left[
\begin{array}{cccccc}
w_{rt}^{k}+x\ g_{k} & -xv_{r}^{k-1}\left( -\left( -q\right)
^{r}\right) \binom{k}{1} & ... &
-xv_{r}\left( -\left( -q\right) ^{r}\right) ^{k-1}\binom{k}{k-1} \\
x\ g_{k-1} & 1-xv_{r}^{k-2}\left( -\left( -q\right) ^{r}\right) \binom{k-1%
}{1} & ...
 &
-x\left( -\left( -q\right) ^{r}\right) ^{k-1}\binom{k-1}{k-1} \\
x\ g_{k-2}  & -xv_{r}^{k-3}\left( -\left( -q\right) ^{r}\right) \binom{k-2%
}{1}&  ...  &0 & \\
... & ... & ... & ...  \\
x\ g_{2}  & -xv_{r}\left( -\left( -q\right) ^{r}\right)
\binom{2}{1}& ... &0 &
   \\
x\ g_{1} & -x\left( -\left( -q\right) ^{r}\right) \binom{1}{1}&...
&1&
%
\end{array}%
\right],
\end{eqnarray*}%
where \ $g_{j}=\left( w_{r\left( t+1\right)
}^{j}-v_{r}^{j}w_{rt}^{j}\right) w_{rt}^{k-j}$ for all $j=1,2,...k.$

Stanica \cite{7} found the generating function of powers of terms of $%
\left\{ W_{n}\right\} $ given by \cite{1},
$\sum\limits_{n=0}^{\infty }W_{n}^{k}x^{n}.$ Considering Stanica's
result, we give the following result for the generating function
\begin{equation*}
\Re _{k,t,r}\left( x\right) =\sum\limits_{n=t}^{\infty
}W_{rn}^{k}x^{n}
\end{equation*}
as the following Lemma~\ref{lemma2}.

\begin{lemma}
For odd $k,$%
\begin{eqnarray*}
\Re _{k,t,r}\left( x\right) &=&\frac{1}{\left( \alpha -\beta \right) ^{k}}%
\sum\limits_{j=0}^{\frac{k-1}{2}}\left( -AB\right) ^{j}\binom{k}{j} \\
&&\times \frac{A^{k-2j}-B^{k-2j}+\left( -q\right) ^{rj}\left(
B^{k-2j}\alpha
^{r\left( k-2j\right) }-A^{k-2j}\beta ^{r\left( k-2j\right) }\right) x}{%
1-\left( -q\right) ^{rj}V_{r\left( k-2j\right) }x-q^{rk}x^{2}} \\
&&-\sum\limits_{n=0}^{t-1}W_{rn}^{k}x^{n}
\end{eqnarray*}%
and for even $k,$%
\begin{eqnarray*}
\Re _{k,t,r}\left( x\right) &=&\frac{1}{\left( \alpha -\beta \right) ^{k}}%
\sum\limits_{j=0}^{\frac{k}{2}-1}\left( -AB\right) ^{j}\binom{k}{j} \\
&&\times \frac{A^{k-2j}+B^{k-2j}-\left( -q\right) ^{rj}\left(
B^{k-2j}\alpha
^{r\left( k-2j\right) }+A^{k-2j}\beta ^{r\left( k-2j\right) }\right) x}{%
1-\left( -q\right) ^{rj}V_{r\left( k-2j\right) }x+q^{rk}x^{2}} \\
&&+\binom{k}{k/2}\frac{\left( -AB\right) ^{k/2}}{1-\left( -q\right) ^{k/2}x}%
-\sum\limits_{n=0}^{t-1}W_{rn}^{k}x^{n}.
\end{eqnarray*}
\label{lemma2}
\end{lemma}

\begin{proof}
The proof easily follows from \cite{7}.
\end{proof}

For further use, we define a family $\left\{ A_{k,d,t,r}\right\}
_{d=1}^{k}$
of generating functions by%
\begin{equation}
A_{k,d,t,r}\left( x\right) =\sum_{n=t}^{\infty
}W_{rn}^{k-d}W_{r\left( n+1\right) }^{d}x^{n+1}.  
\end{equation}%
Now, we give two relations between the generating functions $%
A_{k,d,t,r}\left( x\right) $ and $\Re _{k,t,r}\left( x\right) $.

\begin{lemma}
For $k\geq 1$, positive integer $r$ and non-negative integer $t,$%
\begin{eqnarray*}
\left( 1-V_{r}^{k}x+\left( -\left( -q\right) ^{r}\right)
^{k}x^{2}\right) \Re _{k,t,r}\left( x\right)
-x\sum\limits_{j=1}^{k-1}\binom{k}{j}\left(
-\left( -q\right) ^{r}\right) ^{j}V_{r}^{k-j}A_{k,k-j,t,r}\left( x\right) \\
=W_{rt}^{k}x^{t}+\left( W_{r\left( t+1\right)
}^{k}-V_{r}^{k}W_{rt}^{k}\right) x^{t+1}.
\end{eqnarray*}
\label{lemma3}
\end{lemma}

\begin{proof}
Using the binomial theorem, we get%
\begin{eqnarray*}
W_{r\left( n+2\right) }^{k} &=&\left( V_{r}W_{r\left( n+1\right)
}-\left(
-q\right) ^{r}W_{rn}\right) ^{k} \\
&=&V_{r}^{k}W_{r\left( n+1\right) }^{k}+\sum\limits_{j=1}^{k-1}\binom{k}{j}%
\left( -\left( -q\right) ^{r}\right) ^{j}V_{r}^{k-j}W_{r\left(
n+1\right) }^{k-j}W_{rn}^{j}+\left( -\left( -q\right) ^{r}\right)
^{k}W_{rn}^{k}.
\end{eqnarray*}%
Multiplying by $x^{n+2}$ and summing over all $n\geq t$, using definition \cite{2}, we get
\begin{eqnarray*}
x^{n+2}W_{r\left( n+2\right) }^{k} &=&x^{n+2}V_{r}^{k}W_{r\left(
n+1\right) }^{k}+x^{n+2}\sum\limits_{j=1}^{k-1}\binom{k}{j}\left(
-\left( -q\right)
^{r}\right) ^{j}V_{r}^{k-j}W_{r\left( n+1\right) }^{k-j}W_{rn}^{j} \\
&&+x^{n+2}\left( -\left( -q\right) ^{r}\right) ^{k}W_{rn}^{k}
\end{eqnarray*}%
and so%
\begin{eqnarray*}
&&\Re _{k,t,r}\left( x\right) -W_{rt}^{k}x^{t}-W_{r\left( t+1\right)
}^{k}x^{t+1} \\
&=&xV_{r}^{k}\Re _{k,t,r}\left( x\right) -x^{t+1}V_{r}^{k}W_{rt}^{k} \\
&&+x\sum\limits_{j=1}^{k-1}\binom{k}{j}\left( -\left( -q\right)
^{r}\right) ^{j}V_{r}^{k-j}A_{k,k-j,r}\left( x\right) +x^{2}\left(
-\left( -q\right) ^{r}\right) ^{k}\Re _{k,t,r}\left( x\right) ,
\end{eqnarray*}%
which, by a simple arrangement, completes the proof.
\end{proof}

\begin{lemma}
For any $k\geq 1$, positive integer $r$, non-negative integer $t,$ and $%
d\geq t+1,$%
\begin{eqnarray*}
A_{k,d,t,r}\left( x\right) -x^{t+1}W_{rt}^{k-d}W_{r\left( t+1\right)
}^{d}
&=&xV_{r}^{d}\left( \Re _{k,t,r}\left( x\right) -x^{t}W_{rt}^{k}\right) \\
&&+x\sum\limits_{j=1}^{d}\binom{d}{j}\left( -\left( -q\right)
^{r}\right) ^{j}V_{r}^{d-j}A_{k,k-j,t,r}\left( x\right) .
\end{eqnarray*}
\end{lemma}

\begin{proof}
Using the binomial theorem, we have
\begin{eqnarray*}
W_{rn}^{k-d}W_{r\left( n+1\right) }^{d} &=&W_{rn}^{k-d}\left(
V_{r}W_{rn}-\left( -q\right) ^{r}W_{r\left( n-1\right) }\right) ^{d} \\
&=&W_{rn}^{k-d}\sum\limits_{j=0}^{d}\binom{d}{j}V_{r}^{d-j}\left(
-\left( -q\right) ^{r}\right) ^{j}W_{rn}^{d-j}W_{r\left( n-1\right)
}^{j}.
\end{eqnarray*}%
Multiplying by $x^{n+1}$ and summing over all $n\geq t+1$, we obtain
the
claimed result:%
\begin{multline*}
A_{k,d,t,r}\left( x\right) -x^{t+1}W_{rt}^{k-d}W_{r\left( t+1\right)
}^{d}=xV_{r}^{d}\left( \Re _{k,t,r}\left( x\right) -x^{t}W_{rt}^{k}\right) \\
+x\sum\limits_{j=1}^{d}\binom{d}{j}\left( -\left( -q\right)
^{r}\right) ^{j}V_{r}^{d-j}A_{k,k-j,t,r}\left( x\right) .
\end{multline*}
\end{proof}

Now, we shall mention our main result:

\begin{theorem}
For any $k\geq 1$, positive integer $r$, non-negative
integer $t,$ the generating function $\Re _{k,t,r}\left( x\right) $
is
\begin{equation}
\frac{\det \left( \delta _{k,t,r}\right) }{\det \left( \Delta _{k,r}\right) }%
.
\end{equation}
\label{thm4}
\end{theorem}

\begin{proof}
By using Lemma~\ref{lemma2} and Lemma~\ref{lemma3}, we obtain%
\begin{equation*}
\Delta _{k,r}\left[ \Re _{k,t,r}\left( x\right) ,A_{k,k-1,t,r}\left(
x\right) ,A_{k,k-2,t,r}\left( x\right) ,...A_{k,1,t,r}\left(
x\right) \right] ^{T}=\upsilon _{k,t,r},
\end{equation*}%
where $\upsilon _{k,t,r}$ is given by%
\begin{eqnarray*}
&&\left[ W_{rt}^{k}x^{t}+\left( W_{r\left( t+1\right)
}^{k}-V_{r}^{k}W_{rt}^{k}\right) x^{t+1},\left( W_{rt}W_{r\left(
t+1\right)
}^{k-1}-V_{r}^{k-1}W_{rt}^{k}\right) x^{t+1},\right. \\
&&\left. \left( W_{rt}^{2}W_{r\left( t+1\right)
}^{k-2}-V_{r}^{k-2}W_{rt}^{k}\right) x^{t+1},...,\left(
W_{rt}^{k-1}W_{r\left( t+1\right) }-V_{r}W_{rt}^{k}\right)
x^{t+1}\right] .
\end{eqnarray*}%
Hence the solution of the above equation gives the generating
function\\ $\Re _{k,t,r}\left( x\right) =\left( \det \left( \delta
_{k,t,r}\right) \right) /\left( \det \left( \Delta _{k,r}\right)
\right) .$
\end{proof}

\section{Applications}

We state some applications of our main result by the following tables:

\begin{table}[H]
\begin{center}
\caption{The generating function for the powers of Fibonacci numbers}
\ \\
\begin{tabular}{|l|l|l|l|}
\hline $k$ & $t$ & $r$ & $\text{The generating function }\Re
_{k,t,r}\left( x;0,1,1,1\right) $ \\ \hline\hline $1$ & $1$ & $2$ &
$\frac{1}{\allowbreak 1-3x+x^{2}}$ \\ \hline $2$ & $1$ & $2$ &
$\frac{1+x}{\left( 1-x\right) \left( 1-7x+x^{2}\right) }$
\\ \hline
$3$ & $1$ & $2$ & $\frac{1+6x+x^{2}}{1-21x+56x^{2}-21x^{3}+x^{4}}$
\\ \hline $4$ & $1$ & $2$ &
$\frac{16+1712x+1712x^{2}+17x^{3}}{\left( 1-x\right) \left(
1-34x+x^{2}\right) \left( 1-1154x+x^{2}\right) }$ \\ \hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[H]
\begin{center}
\caption{The generating function for the powers of Lucas numbers}
\ \\
\begin{tabular}{|l|l|l|l|}
\hline $k$ & $t$ & $r$ & $\text{The generating function }\Re
_{k,t,r}\left( x;2,1,1,1\right) $ \\ \hline\hline $1$ & $1$ & $2$ &
$\frac{3-2x}{\allowbreak 1-3x+x^{2}}$ \\ \hline $2$ & $1$ & $2$ &
$\frac{9-23x+4x^{2}}{\left( 1-x\right) \left( 1-7x+x^{2}\right) }$
\\ \hline
$3$ & $1$ & $2$ & $\frac{27-224x+141x^{2}-8x^{3}}{1-21x+56x^{2}-21x^{3}+x^{4}%
}$ \\ \hline
$4$ & $1$ & $2$ & $\frac{81-2054x+452913226x^{2}-78298x^{3}-2864x^{4}}{%
\left( 1-x\right) \left( 1-7x+x^{2}\right) \left( 1-47x+x^{2}\right) }$ \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[H]
\begin{center}
\caption{The generating function for the powers of Pell numbers}
\ \\
\begin{tabular}{|l|l|l|l|}
\hline $k$ & $t$ & $r$ & $\text{The generating function }\Re
_{k,t,r}\left( x;0,1,2,1\right) $ \\ \hline\hline $1$ & $1$ & $2$ &
$\frac{2}{x^{2}-6x+1}$ \\ \hline $2$ & $1$ & $2$ &
$\frac{4+4x}{\left( 1-x\right) \left( 1-34x+x^{2}\right) }$
\\ \hline
$3$ & $1$ & $2$ & $\frac{\allowbreak 8\left( 1+12x+x^{2}\right) }{%
1-204x+1190x^{2}-204x^{3}+x^{4}}$ \\ \hline
$4$ & $1$ & $2$ & $\frac{16\left( x+1\right) \left( 1+106x+x^{2}\right) }{%
\left( 1-x\right) \left( 1-34x+x^{2}\right) \left(
1-1154x+x^{2}\right) }$
\\ \hline
\end{tabular}%
\end{center}
\end{table}


\begin{table}[H]
\begin{center}
\caption{The generating function for the powers of Chebyshev
polynomials of the second kind}
\ \\
\begin{tabular}{|l|l|l|l|}
\hline $k$ & $t$ & $r$ & $\text{The generating function }\Re
_{k,t,r}\left( x;1,2t,2t,-1\right) $ \\ \hline\hline $1$ & $1$ & $2$
& $\frac{-1+4t^{2}-x}{1+\left( \allowbreak 2-4t^{2}\right) x+x^{2}}$
\\ \hline $2$ & $1$ & $2$ & $\frac{\left( 16t^{4}-8t^{2}+1\right)
+\left( 16t^{2}-16t^{4}-2\right) x+x^{2}}{\left( 1-x\right) \left(
1+\left( -2+12t^{2}\right) x+x^{2}\right) }$ \\ \hline $3$ & $1$ &
$2$ & $\frac{12t^{2}-48t^{4}+64t^{6}+27-\left(
4-24t^{2}+288t^{4}-256t^{6}+576t^{8}\right) x+\left(
40t^{2}-336t^{4}+64t^{6}-3\right) x^{2}-x^{3}\allowbreak }{1+\left(
-64t^{6}+96t^{4}-\allowbreak 40t^{2}+4\right) x+\left(
256t^{8}-512t^{6}+336t^{4}-80t^{2}+6\right) x^{2}+\left(
-64t^{6}+96\allowbreak t^{4}-40t^{2}+4\right) x^{3}+x^{4}}$ \\
\hline
\end{tabular}
\end{center}
\end{table}

\textbf{Fibonacci numbers.} If $a=0$ and $p=q=b=1,$ then 
Theorem~\ref{thm4} for $k=1,2,3,4$ yields Table 1.

\textbf{Lucas numbers.} If $a=2$ and $p=q=b=1,$ then Theorem~\ref{thm4} for $%
k=1,2,3,4$ yields Table 2.

\textbf{Pell numbers.} If $a=0$ and $p=2,q=b=1,$ then Theorem~\ref{thm4} for $%
k=1,2,3,4$ yields Table 3.

\textbf{Chebyshev polynomials of the second kind.} If $a=1$, $b=p=2t$ and $%
q=-1,$ then Theorem~\ref{thm4} for $k=1,2,3$ yields Table 4.

Applying Theorem~\ref{thm4} for $k=1,2,3$, then we give the following
corollary.

\begin{corollary}
Let $k=1,2,3$. Then the generating function $\Re _{k,t,r}\left(
x;a,b,p,q\right) $ is given by $\hat{A}_{k,t,r}\left( x\right) /\hat{E}%
_{k,t,r}\left( x\right) ,$ where
\begin{eqnarray*}
\hat{A}_{1,1,2}\left( x\right) &=&aq+bp-aq^{2}x, \\
\hat{A}_{2,1,2}\left( x\right)
&=&a^{2}q^{2}+b^{2}p^{2}+2abpq+q^{2}\left(
-2a^{2}q^{2}+b^{2}p^{2}-2abp^{3}-2a^{2}p^{2}q-2abpq\right) x \\
&&+a^{2}q^{6}x^{2}, \\
\hat{A}_{3,1,2}\left( x\right)
&=&b^{3}p^{3}+3ab^{2}p^{2}q+3a^{2}bpq^{2}+a^{3}q^{3}-(3a^{3}p^{4}q^{4}+7%
\allowbreak a^{3}p^{2}q^{5} \\
&&+3a^{3}q^{6}+6a^{2}bp^{5}q^{3}+15a^{2}bp^{3}q^{4}+6a^{2}bpq^{5}+6ab^{2}p^{4}q^{3}
\\
&&-2b^{3}p^{5}q^{2}-4b^{3}p^{3}q^{3}+3ab^{2}p^{6}q^{2})x+(3a^{3}p^{4}q^{7}+7a^{3}p^{2}q^{8}+3a^{3}q^{9}
\\
&&+3a^{2}bp^{5}q^{6}+6a^{2}bp^{3}q^{7}+3a^{2}bpq^{8}-3ab^{2}p^{4}q^{6} \\
&&-3ab^{2}p^{2}q^{7}+b^{3}p^{3}q^{6})x^{2}-a^{3}q^{12}x^{3}
\end{eqnarray*}%
\ \ and%
\begin{eqnarray*}
\hat{E}_{1,1,2}\left( x\right) &=&1-\left( p^{2}+2q\right) x+q^{2}x^{2}, \\
\hat{E}_{2,1,2}\left( x\right) &=&\left( q^{2}x-1\right) \left(
-1+\left(
p^{4}+4p^{2}q+2q^{2}\right) x-q^{4}x^{2}\right) , \\
\hat{E}_{3,1,2}\left( x\right) &=&(-1+q^{2}\left( 12q+p^{2}\right)
x-q^{6}x^{2}) \\
&&\times \left( -1+\left( 2q+p^{2}\right)
(4p^{2}q+p^{4}+q^{2}x-q^{6}x^{2}\right) .
\end{eqnarray*}
\end{corollary}

\begin{thebibliography}{9}
\bibitem{1} A. F. Horadam, Basic properties of a certain generalized sequence
of numbers,\textit{Fibonacci Quart.} \textbf{3} (1965),
161--176.

\bibitem{2} A.F. Horadam, Generating functions for powers of a certain
generalized sequence of numbers, \textit{Duke Math. J.} \textbf{32}
(1965), 437--446.

\bibitem{3} T. Mansour, A formula for the generating functions of powers of
Horadam's sequence, \textit{Australasian J. Combinatorics} \textbf{30}
(2004), 207--212.

\bibitem{4} J. Riordan, Generating function for powers of Fibonacci numbers,
\textit{Duke Math. J.} \textbf{29} (1962), 5--12.

\bibitem{5} P. Haukkanen and J. Rutkowski, On generating functions for
powers of recurrence sequences, \textit{Fibonacci Quart.}
\textbf{29} (1991), 329--332.

\bibitem{6} P. Haukkanen, A note on Horadam's sequence, \textit{Fibonacci
Quart.}, \textbf{40} (2002), 358--361.

\bibitem{7} P. Stanica, Generating functions, weighted and non-weighted
sums for powers of second-order recurrence sequences, \textit{Fibonacci
Quart.} \textbf{41} (2003), 321--333.

\bibitem{8} E. K\i l\i \c{c} and P. Stanica, Factorizations of binary
polynomial recurrences by matrix methods, \textit{Rocky Mount. J.
Math.}, to appear.

\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B37; Secondary 11B39, 05A15.

\noindent \emph{Keywords: } second-order linear recurrence,
generating function.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A000032},
\seqnum{A000045}, and
\seqnum{A000129}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received December 6 2010;
revised version received January 27 2011; May 3 2011. 
Published in {\it Journal of Integer Sequences}, May 3 2011.

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\noindent
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