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\theoremstyle{plain}
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\begin{center}
\vskip1cm{\LARGE \textbf{Generalized Bivariate Lucas $p$-Polynomials\\
\vskip.1in
and Hessenberg Matrices }} 
{\large
\vskip1cm 
Kenan Kaygisiz and Adem \c{S}ahin\\ 
Department of Mathematics \\
Faculty of Arts and Sciences\\ 
Gaziosmanpa\c{s}a University\\
60250 Tokat \\
Turkey \\
\href{mailto:kenan.kaygisiz@gop.edu.tr}{\texttt{kenan.kaygisiz@gop.edu.tr}}%
\\[0pt]
\href{mailto:adem.sahin@gop.edu.tr}{\texttt{adem.sahin@gop.edu.tr}}\\[0pt]
}
\end{center}


\begin{abstract}
In this paper, we give some determinantal and permanental
representations of generalized bivariate Lucas $p$-polynomials by
using various Hessenberg matrices. The results that we obtained are
important since generalized bivariate Lucas $p$-polynomials are
general forms of, for example, bivariate Jacobsthal-Lucas, bivariate
Pell-Lucas $p$-polynomials, Chebyshev polynomials of the first kind,
Jacobsthal-Lucas numbers etc.
\end{abstract}

\section{Introduction}

The generalized Lucas $p$-numbers \cite{stakhov} are defined by

\begin{equation}
L_{p}(n)=L_{p}(n-1)+L_{p}(n-p-1)\
\end{equation}%
for $n>p+1$, with boundary conditions $L_{p}(0)=(p+1),$ $L_{p}(1)=\cdots
=L_{p}(p)=1.$

The Lucas \cite{alexlupa}, Pell-Lucas \cite{horadam} and Chebyshev
polynomials of the first kind
\cite{udrea} 
are defined as follows:
\begin{eqnarray*}
l_{n+1}(x) &=&xl_{n}(x)+l_{n-1}(x),\text{ }n\geq 2\text{ with }l_{0}(x)=2,%
\text{ }l_{1}(x)=x \\
Q_{n+1}(x) &=&2xQ_{n}(x)+Q_{n-1}(x),\text{ }n\geq 2\text{ with }Q_{0}(x)=2,%
\text{ }Q_{1}(x)=2x \\
T_{n+1}(x) &=&2xT_{n}(x)-T_{n-1}(x),\text{ }n\geq 2\text{ with }T_{0}(x)=1,%
\text{ }T_{1}(x)=x
\end{eqnarray*}%
respectively.

The generalized bivariate Lucas $p$-polynomials \cite{tuglu} are defined 
as follows:
\begin{equation*}
L_{p,n}(x,y)=xL_{p,n-1}(x,y)+yL_{p,n-p-1}(x,y)
\end{equation*}%
for $n>p$, with boundary conditions $L_{p,0}(x,y)=(p+1),$ $%
L_{p,n}(x,y)=x^{n},$ $n=1,2,\ldots,p.$

A few terms of $L_{p,n}(x,y)$ for $p=4$ and $p=5$ are

$%
5,x,x^{2},x^{3},x^{4},5y+x^{5},6xy+x^{6},x^{7}+7x^{2}y,x^{8}+8x^{3}y,x^{9}+9x^{4}y,5y^{2}+x^{10}+10x^{5}y,\ldots
$
and
$6,x,x^{2},x^{3},x^{4},x^{5},5y+x^{6},6xy+x^{7},x^{8}+7x^{2}y,x^{9}+8x^{3}y,%
\ldots $
respectively.

MacHenry \cite{mach1} defined generalized Lucas polynomials $(L_{k,n}(t))$
where $t_{i}$ $(1\leq i\leq k)$ are constant coefficients of the core
polynomial%
\begin{equation*}
P(x;t_{1},t_{2},\ldots ,t_{k})=x^{k}-t_{1}x^{k-1}-\cdots -t_{k},
\end{equation*}%
which is denoted by the vector $t=(t_{1},t_{2},\ldots ,t_{k}).$

$G_{k,n}(t_{1},t_{2},\ldots ,t_{k})$ is defined by%
\begin{eqnarray}
G_{k,n}(t) &=&0,\text{ }n<0  \notag \\
G_{k,0}(t) &=&\text{ }k  \notag \\
G_{k,1}(t) &=&t_{1}\text{ }  \notag \\
G_{k,n+1}(t) &=&t_{1}G_{k,n}(t)+\cdots +t_{k}G_{k,n-k+1}(t).  \notag
\end{eqnarray}

MacHenry obtained very useful properties of these polynomials in \cite%
{mach2, mach3}.

\begin{remark}
\label{remark} \cite{tuglu}Cognate polynomial sequence are as follows%
\begin{equation*}
\begin{tabular}{llll}
\hline
$\mathbf{x}$ & $\mathbf{y}$ & $\mathbf{p}$ & $L_{p,n}(x,y)$ \\ \hline
$x$ & $y$ & $1$ & bivariate Lucas polynomials $L_{n}(x,y)$ \\
$x$ & $1$ & $p$ & Lucas $p-$polynomials $L_{p,n}(x)$ \\
$x$ & $1$ & $1$ & Lucas polynomials $l_{n}(x)$ \\
$1$ & $1$ & $p$ & Lucas $p-$numbers $L_{p}(n)$ \\
$1$ & $1$ & $1$ & Lucas numbers $L_{n}$ \\
$2x$ & $y$ & $p$ & bivariate Pell-Lucas $p$-polynomials $L_{p,n}(2x,y)$ \\
$2x$ & $y$ & $1$ & bivariate Pell-Lucas polynomials $L_{n}(2x,y)$ \\
$2x$ & $1$ & $p$ & Pell-Lucas $p$-polynomials $Q_{p,n}(x)$ \\
$2x$ & $1$ & $1$ & Pell-Lucas polynomials $Q_{n}(x)$ \\
$2$ & $1$ & $1$ & Pell-Lucas numbers $Q_{n}$ \\
$2x$ & $-1$ & $1$ & Chebyshev polynomials of the first kind $T_{n}(x)$ \\
$x$ & $2y$ & $p$ & bivariate Jacobsthal-Lucas $p$-polynomials $L_{p,n}(x,2y)$
\\
$x$ & $2y$ & $1$ & Bivariate Jacobsthal-Lucas polynomials $L_{n}(x,2y)$ \\
$1$ & $2y$ & $1$ & Jacobsthal-Lucas polynomials $j_{n}(y)$ \\
$1$ & $2$ & $1$ & Jacobsthal-Lucas numbers $j_{n}$ \\ \hline
\end{tabular}%
\end{equation*}
\end{remark}

Remark \ref{remark} shows that $L_{p,n}(x,y)$ is a general form of
all sequences and polynomials mentioned in that remark. Therefore,
any result obtained from $L_{p,n}(x,y)$ is valid for all sequences
and polynomials mentioned there.

Many researchers have studied determinantal and permanental representations of $k$
sequences of generalized order-$k$ Fibonacci and Lucas numbers. For
example, Minc \cite{min} defined an $n\times n$ (0,1)-matrix
$F(n,k),$ and showed that the permanents of $F(n,k)$ are equal to
the generalized order-$k$ Fibonacci numbers. Nalli and Haukkanen
\cite{nal} defined $h(x)$-Fibonacci and Lucas polynomials and gave
determinantal representations of these polynomials. The authors
(\cite{lee,lee2}) defined two $(0,1)$-matrices and showed that the
permanents of these matrices are the generalized Fibonacci and Lucas
numbers. \"{O}cal et al. \cite{oca} gave some determinantal and
permanental representations of $k$-generalized Fibonacci and Lucas
numbers, and obtained Binet's formula for these sequences. K\i l\i c
and Stakhov \cite{kilic2} gave permanent representation of Fibonacci
and Lucas $p$-numbers. K\i l\i c and Tasci \cite{kilic3} studied
permanents and determinants of Hessenberg matrices. Janjic
\cite{jan} considers a particular case of upper Hessenberg matrices
and gave a determinant representation of a generalized Fibonacci
numbers.

In this paper, we give some determinantal and permanental representations of
$L_{p,n}(x,y)$ by using various Hessenberg matrices. These results are a
general form of determinantal and permanental representations of polynomials
and sequences mentioned in Remark \ref{remark}.

\section{The determinantal representations}

In this section, we give some determinantal representations of $L_{p,n}(x,y)$
using Hessenberg matrices. 

\begin{definition}
An $n\times n$ matrix $A_{n}=(a_{ij})$ is called lower Hessenberg matrix if $%
a_{ij}=0$ when $j-i>1$ i.e.,%
\begin{equation*}
A_{n}=\left[
\begin{array}{ccccc}
a_{11} & a_{12} & 0 & \cdots & 0 \\
a_{21} & a_{22} & a_{23} & \cdots & 0 \\
a_{31} & a_{32} & a_{33} & \cdots & 0 \\
\vdots & \vdots & \vdots &  & \vdots \\
a_{n-1,1} & a_{n-1,2} & a_{n-1,3} & \cdots & a_{n-1,n} \\
a_{n,1} & a_{n,2} & a_{n,3} & \cdots & a_{n,n}%
\end{array}%
\right].
\end{equation*}
\end{definition}

\begin{theorem}
\label{cahill} \cite{cah} Let $A_{n}$ be an $n\times n$ lower Hessenberg
matrix for all $n\geq 1$ and $\det (A_{0})=1$. Then,
\begin{equation*}
\det (A_{1})=a_{11}
\end{equation*}%
and for $n\geq 2$
\begin{equation*}
\quad \quad \quad \det (A_{n})=a_{n,n}\det (A_{n-1})+\sum\limits_{r=1}^{n-1}
\left[ (-1)^{n-r}a_{n,r}(\prod\limits_{j=r}^{n-1}a_{j,j+1})\det (A_{r-1})%
\right] .
\end{equation*}
\end{theorem}

\begin{theorem}
\label{t1}Let $L_{p,n}(x,y)$ be the generalized bivariate Lucas $p$%
-polynomials and $W_{p,n}=(w_{ij})$ be an $n\times n$ Hessenberg matrix
defined by
\begin{equation*}
w_{ij}=
\begin{cases}
i, & \text{if \ }i=j-1; \\
x, & \text{if \ }i=j; \\
i^{p}y, & \text{if \ }p=i-j\text{ and }j\neq 1; \\
(p+1)i^{p}y, & \text{if \ }p=i-j\text{ and }j=1; \\
0, & \text{otherwise;}%
\end{cases}%
\end{equation*}%
that is,%
\begin{equation}
W_{p,n}=\left[
\begin{array}{ccccc}
x & i & 0 & \cdots & 0 \\
0 & x & i & \ddots & \vdots \\
\vdots & 0 & x &  & 0 \\
(p+1)i^{p}y & 0 & \vdots & \cdots &  \\
0 & i^{p}y & 0 &  & 0 \\
\vdots & 0 & \ddots & x & i \\
0 & 0 & \cdots & 0 & x%
\end{array}%
\right] .  \label{kuka}
\end{equation}%
Then,%
\begin{equation}
\det (W_{p,n})=L_{p,n}(x,y)  \label{tt1}
\end{equation}%
where $n\geq 1$ and $i=\sqrt{-1}.$
\end{theorem}

\begin{proof}
To prove (\ref{tt1}), we use mathematical induction on $n$. The result is
true for $n=1$ by hypothesis.

Assume that it is true for all positive integers less than or equal to $n,$
namely $\det (W_{p,n})=L_{p,n}(x,y)$. Then, we have
\begin{eqnarray*}
\det (W_{p,n+1}) &=&q_{n+1,n+1}\det (W_{p,n})+\sum\limits_{r=1}^{n}\left[
(-1)^{n+1-r}q_{n+1,r}(\prod\limits_{j=r}^{n}q_{j,j+1})\det (W_{p,r-1})\right]
\\
&=&x\det (W_{p,n})+\sum\limits_{r=1}^{n-p}\left[ (-1)^{n+1-r}q_{n+1,r}(\prod%
\limits_{j=r}^{n}q_{j,j+1})\det (W_{p,r-1})\right] \\
&&+\sum\limits_{r=n-p+1}^{n}\left[ (-1)^{n+1-r}q_{n+1,r}(\prod%
\limits_{j=r}^{n}q_{j,j+1})\det (W_{p,r-1})\right] \\
&=&x\det (W_{p,n})+\left[ (-1)^{p}(i)^{p}y\prod\limits_{j=n-p+1}^{n}i\det
(W_{p,n-p})\right] \\
&=&x\det (W_{p,n})+\left[ (-1)^{p}y(i)^{p}.(i)^{p}\det (W_{p,n-p})\right] \\
&=&x\det (W_{p,n})+y\det (W_{p,n-p})
\end{eqnarray*}%
by using Theorem \ref{cahill}. From the induction hypothesis and the
definition of $L_{p,n}(x,y)$ we obtain
\begin{equation*}
\det (W_{p,n+1})=xL_{p,n}(x,y)+yL_{p,n-p}(x,y)=L_{p,n+1}(x,y).
\end{equation*}%
Therefore, (\ref{tt1}) holds for all positive integers $n$.
\end{proof}

\begin{example}
We obtain the $5$-th $L_{p,n}(x,y)$ for $p=4$, by using Theorem \ref{t1}%
\begin{equation*}
\ L_{4,5}(x,y)=\det \left[
\begin{array}{ccccc}
x & i & 0 & 0 & 0 \\
0 & x & i & 0 & 0 \\
0 & 0 & x & i & 0 \\
0 & 0 & 0 & x & i \\
5i^{4}y & 0 & 0 & 0 & x%
\end{array}%
\right] =5y+x^{5}.
\end{equation*}
\end{example}

\begin{theorem}
\bigskip \label{t2}Let $p\geq 1$ be an integer$,$ $L_{p,n}(x,y)$ be the
generalized bivariate Lucas $p$-polynomials and $M_{p,n}=(m_{ij})$ be an $%
n\times n$ Hessenberg matrix defined by
\begin{equation*}
m_{ij}=
\begin{cases}
-1, & \text{if \ }j=i+1; \\
x, & \text{if\ \ }i=j; \\
y, & \text{if \ }p=i-j\text{ and }j\neq 1; \\
(p+1)y, & \text{if \ }p=i-j\text{ and }j=1; \\
0,\text{\ } & \text{otherwise;}%
\end{cases}%
\end{equation*}%
that is,
\begin{equation}
M_{p,n}=\left[
\begin{array}{ccccc}
x & -1 & 0 & \cdots & 0 \\
0 & x & -1 & \cdots & 0 \\
0 & 0 & x & \cdots & 0 \\
\vdots & \vdots & \vdots &  & \vdots \\
(p+1)y & 0 & 0 & \cdots & 0 \\
0 & y & 0 & \cdots & 0 \\
& \vdots & \vdots & \ddots & -1 \\
0 & 0 & \cdots & 0 & x%
\end{array}%
\right] .  \label{beka}
\end{equation}%
Then,%
\begin{equation*}
\det (M_{p,n})=L_{p,n}(x,y).
\end{equation*}
\end{theorem}

\begin{proof}
Since the proof is similar to the proof of Theorem \ref{t1}, we omit the
details.
\end{proof}

\section{The permanent representations}

Let $A=(a_{i,j})$ be a square matrix of order $n$ over a ring R. The
permanent of $A$ is defined by%
\begin{equation*}
\text{per}(A)=\sum\limits_{\sigma \in
S_{n}}\prod\limits_{i=1}^{n}a_{i,\sigma (i)}
\end{equation*}%
where $S_{n}$ denotes the symmetric group on $n$ letters.

\begin{theorem}
\label{ocal} \cite{oca} Let $A_{n}$ be an $n\times n$ lower Hessenberg
matrix for all $n\geq 1$ and per$(A_{0})=1.$ Then, $\text{per}(A_{1})=a_{11}$
and for $n\geq 2$,
\begin{equation*}
\text{per}(A_{n})=a_{n,n}\text{per}(A_{n-1})+\sum\limits_{r=1}^{n-1}\left[
a_{n,r}(\prod\limits_{j=r}^{n-1}a_{j,j+1})\text{per}(A_{r-1})\right] .
\end{equation*}
\end{theorem}

\begin{theorem}
\bigskip \bigskip \label{t3}Let $p\geq 1$ be an integer$,$ $L_{p,n}(x,y)$ be
the generalized bivariate Lucas $p$-polynomials and $H_{p,n}=(h_{rs})$ be an
$n\times n$ lower Hessenberg matrix such that%
\begin{equation*}
h_{rs}=
\begin{cases}
-i, & \text{if \ }s-r=1\text{ }; \\
x, & \text{if \ }r=s\text{ }; \\
i^{p}y, & \text{if \ }p=r-s\text{ and }s\neq 1,\text{ }; \\
(p+1)i^{p}y, & \text{if \ }p=r-s\text{ and }s=1; \\
0, & \text{otherwise;}%
\end{cases}%
\end{equation*}%
then
\begin{equation*}
\text{per}(H_{p,n})=L_{p,n}(x,y)
\end{equation*}%
where $n\geq 1$ and $i=\sqrt{-1}.$
\end{theorem}

\begin{proof}
This is similar to the proof of Theorem \ref{t1} using Theorem \ref{ocal}.
\end{proof}

\begin{example}
We obtain the $6$-th $L_{p,n}(x,y)$ for $p=4$, by using Theorem \ref{t3}%
\begin{equation*}
L_{4,6}(x,y)=\text{per}\left[
\begin{array}{cccccc}
x & -i & 0 & 0 & 0 & 0 \\
0 & x & -i & 0 & 0 & 0 \\
0 & 0 & x & -i & 0 & 0 \\
0 & 0 & 0 & x & -i & 0 \\
5y & 0 & 0 & 0 & x & -i \\
0 & y & 0 & 0 & 0 & x%
\end{array}%
\right] =\allowbreak 6xy+x^{6}.
\end{equation*}
\end{example}

\begin{theorem}
\label{t4}Let $p\geq 1$ be an integer$,$ $L_{p,n}(x,y)$ be the generalized
bivariate Lucas $p$-polynomials and $K_{p,n}=(k_{ij})$ be an $n\times n$
lower Hessenberg matrix such that%
\begin{equation*}
k_{ij}=
\begin{cases}
1, & \text{if \ }j=i+1; \\
x, & \text{if \ }i=j; \\
y, & \text{if \ }i-j=p\text{ and }j\neq 1; \\
(p+1)y, & \text{if \ }i-j=p\text{ and }j=1; \\
0,\text{\ } & \text{otherwise;}%
\end{cases}%
\end{equation*}%
then%
\begin{equation*}
\text{per}(K_{p,n})=L_{p,n}(x,y).
\end{equation*}
\end{theorem}

\begin{proof}
This is similar to the proof of Theorem \ref{t1} by using Theorem \ref{ocal}.
\end{proof}

We note that the theorems given above are still valid for the sequences and
polynomials mentioned in Remark \ref{remark}.

\begin{corollary}
If we rewrite Theorem \ref{t1}, Theorem \ref{t2}, Theorem \ref{t3} and
Theorem \ref{t4} for $x,y,p$, we obtain the following table.%
\begin{equation*}
\begin{tabular}{llll}
\hline
\textbf{For}$\ \ \mathbf{x}$ & $\mathbf{y}$ & $\mathbf{p}$ & \ \ \ $\det
(W_{p,n})=\det (M_{p,n})=$per$(H_{p,n})=$per$(K_{p,n})=\mathbf{L}_{p,n+1}%
\mathbf{(x,y)},$ \\ \hline
for$\ \ x$ & $y$ & $1$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})= $per$(K_{p,n})=\mathbf{L}_{n}\mathbf{(x,y)},$ \\
for $\ x$ & $1$ & $p$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$(H_{p,n})=$%
per$(K_{p,n})=$ $\mathbf{L}_{p,n}\mathbf{(x)},$ \\
for$\ \ x$ & $1$ & $1$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})= $per$(K_{p,n})=$\textbf{\ }$l_{n}(x),$ \\
for$\ \ 1$ & $1$ & $p$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})= $per$(K_{p,n})=$ $\mathbf{L}_{p}\mathbf{(n)},$ \\
for$\ \ 1$ & $1$ & $1$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})= $per$(K_{p,n})=$ $\mathbf{L}_{n},$ \\
for$\ \ 2x$ & $y$ & $p$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$\ per$%
(H_{p,n})=$per$(K_{p,n})=$ $\mathbf{L}_{p,n}\mathbf{(2x,y)},$ \\
for$\ \ 2x$ & $y$ & $1$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})=$per$(K_{p,n})=$ $\mathbf{L}_{n}\mathbf{(2x,y)},$ \\
for$\ \ 2x$ & $1$ & $p$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})=$per$(K_{p,n})=$ $\mathbf{Q}_{p,n}\mathbf{(x)},$ \\
for$\ \ 2x$ & $1$ & $1$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})=$per$(K_{p,n})=$ $\mathbf{Q}_{n}\mathbf{(x)},$ \\
for$\ \ 2$ & $1$ & $1$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})= $per$(K_{p,n})=$ $\mathbf{Q}_{n},$ \\
for$\ \ 2x$ & $-1$ & $1$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})=$per$(K_{p,n})=\mathbf{T}_{n}\mathbf{(x)},$ \\
for$\ \ x$ & $2y$ & $p$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})=$per$(K_{p,n})=$ $\mathbf{L}_{p,n}\mathbf{(x,2y)},$ \\
for$\ \ x$ & $2y$ & $1$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})=$per$(K_{p,n})=$ $\mathbf{L}_{n}\mathbf{(x,2y)},$ \\
for$\ \ 1$ & $2y$ & $1$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})=$per$(K_{p,n})=$ $\mathbf{j}_{n}\mathbf{(y)},$ \\
for$\ \ 1$ & $2$ & $1$ & \ \ \ $\det (W_{p,n})=\det (M_{p,n})=$per$%
(H_{p,n})= $per$(K_{p,n})=$\textbf{\ }$j_{n}.$ \\ \hline
\end{tabular}%
\end{equation*}
\end{corollary}

\section{Conclusion}
In this paper, we have given some determinantal and permanental
representations of generalized bivariate Lucas $p$-polynomials. Our results
allow us to derive determinantal and permanantel representations of sequences
and polynomials mentioned in Remark \ref{remark}.

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\end{thebibliography}

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\noindent 2010 \textit{Mathematics Subject Classification}: Primary 11B37,
Secondary 15A15, 15A51.

\noindent \emph{Keywords: } Generalized bivariate Lucas $p$-polynomials,
determinant, permanent and Hessenberg matrix.

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\noindent
Received  November 30 2011;
revised version received  February 21 2012.
Published in {\it Journal of Integer Sequences}, March 11 2012.

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