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\begin{center}
\vskip 1cm{\LARGE \textbf{Some Identities for Fibonacci and Incomplete \\
\vskip .05in
Fibonacci $p$-Numbers via the \\
\vskip .12in
Symmetric Matrix Method }} \vskip 1cm
{\large M. Cetin Firengiz\\[0pt]
Department of Mathematics\\[0pt]
Faculty of Education\\[0pt]
University of Baskent\\[0pt]
Baglica, Ankara\\[0pt]
Turkey\\[0pt]
\href{mailto:mcetin@baskent.edu.tr}{\texttt{mcetin@baskent.edu.tr}}\\[0pt]
\ \\[0pt]
Dursun Tasci and Naim Tuglu\\[0pt]
Department of Mathematics\\[0pt]
University of Gazi\\[0pt]
Teknikokullar, 06500, Ankara\\[0pt]
Turkey \\[0pt]
\href{mailto:dtasci@gazi.edu.tr}{\texttt{dtasci@gazi.edu.tr}} \\[0pt]
\href{mailto:naimtuglu@gazi.edu.tr}{\texttt{naimtuglu@gazi.edu.tr}} \\[0pt]
}
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\begin{abstract}
We obtain some new formulas for the Fibonacci and Lucas
$p$-numbers, by using the symmetric infinite matrix method. We also give
some results for the Fibonacci and Lucas $p$-numbers by means of
the binomial inverse pairing.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}
Dil and Mez\H{o} \cite{3} defined the symmetric
infinite matrix method. For sequences $\left( a_{n}\right) $ and $\left( a^{n}\right) $, the
recursive formula
\begin{eqnarray}
a_{n}^{0} &amp;=&amp;a_{n},\ \ a_{0}^{n}=a^{n}\ \ \ \ \left( n\geq 0\right)  \notag
\\
a_{n}^{k} &amp;=&amp;a_{n-1}^{k}+a_{n}^{k-1}\ \ \ \ \ \ \ \left( n\geq 1,\ k\geq
1\right)  \label{4}
\end{eqnarray}
gives the associated symmetric infinite matrix \cite{3}:
\begin{equation*}
\left(
\begin{array}{cccccccc}
. &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . \\
. &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . \\
. &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . \\
. &amp; . &amp; . &amp;
\begin{array}{c}
a_{n}^{k-1} \\
\downarrow%
\end{array}
&amp; . &amp; . &amp; . &amp; . \\
. &amp; . &amp;
\begin{tabular}{ll}
$a_{n-1}^{k}$ &amp; $\rightarrow $%
\end{tabular}
&amp; a_{n}^{k} &amp; . &amp; . &amp; . &amp; . \\
. &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . \\
. &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . \\
. &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . &amp; .%
\end{array}
\right) .
\end{equation*}

\begin{proposition}
\cite{3} If the relation (\ref{4}) holds, the entry $a_n^k$ of the
corresponding symmetric infinite matrix is
    \begin{equation}
        a_{n}^{k}=\sum\limits_{i=1}^{k}\binom{n+k-i-1}{n-1}a_{0}^{i}+\sum
        \limits_{j=1}^{n}\binom{k+n-j-1}{k-1}a_{j}^{0}.  \label{5}
    \end{equation}
\end{proposition}

For two sequences $(a_{n})$ and $(b_{n})$, the well-known binomial
inverse pair \cite{10} is given by the relations
\begin{equation}
b_{n}=\sum\limits_{k=0}^{n}\binom{n}{k}a_{k}  \label{2}
\end{equation}
\begin{equation}
a_{n}=\sum\limits_{k=0}^{n}\binom{n}{k}\left( -1\right) ^{n-k}b_{k}.
\label{3}
\end{equation}

Stakhov and Rozin \cite{7} defined the Fibonacci $p$-numbers $F_{p}\left(
n\right) $ by the following recurrence relation for $n&gt;p$
\begin{equation}
F_{p}\left( n\right) =F_{p}\left( n-1\right) +F_{p}\left( n-p-1\right)
\label{6}
\end{equation}
with initial conditions
\begin{equation*}
F_{p}\left( 0\right) =0,\ F_{p}\left( n\right) =1\ \ \ \left( n=1,2,\ldots
,p\right)
\end{equation*}
and the Lucas $p$-numbers $L_{p}\left( n\right) $ by the
following recurrence relation for $n&gt;p$
\begin{equation}
L_{p}\left( n\right) =L_{p}\left( n-1\right) +L_{p}\left( n-p-1\right)
\label{7}
\end{equation}
with initial conditions
\begin{equation*}
L_{p}\left( 0\right) =p+1,\ L_{p}\left( n\right) =1\ \ \ \left( n=1,2,\ldots
, p\right) .
\end{equation*}

Note that for the case $p=1$, the sequences of Fibonacci and Lucas $p$-numbers 
reduce to the
well-known Fibonacci and Lucas sequences $\left\{ F_{n}\right\} ,\ \left\{
L_n\right\} $, respectively.
See \cite{1,5,9} for 
more details about the Fibonacci and Lucas $p$-numbers.

Tasci and Cetin-Firengiz \cite{8} introduced the incomplete Fibonacci and Lucas $p$-numbers.
The incomplete Fibonacci $p$-numbers $F_{p}^{k}\left( n\right) $ and the
incomplete Lucas $p$-numbers $L_{p}^{k}\left( n\right) $ are defined by
\begin{equation*}
F_{p}^{k}\left( n\right) =\sum\limits_{j=0}^{k}\binom{n-jp-1}{j}\ \ \ \
\left( n=1,2,\ldots ;\ 0\leq k\leq \left\lfloor \tfrac{n-1}{p+1}%
\right\rfloor \right)
\end{equation*}
and
\begin{equation*}
L_{p}^{k}\left( n\right) =\sum\limits_{j=0}^{k}\frac{n}{n-jp}\binom{n-jp}{j }%
\ \ \ \ \left( n=1,2,\ldots ;\ 0\leq k\leq \left\lfloor \tfrac{n}{p+1}
\right\rfloor \right) .
\end{equation*}
We note that $F_{1}^{\left\lfloor \frac{n-1}{2}\right\rfloor }\left(
n\right) =F_{n},\ L_{1}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(
n\right) =L_{n}$ and $F_{1}^{k}\left( n\right) =F_{n}\left( k\right) $,\ $%
L_{1}^{k}\left( n\right) =L_{n}\left( k\right) $, where $\left\{F_{n}\left(
k\right)\right\} $ and $\left\{L_{n}\left( k\right)\right\} $ are the sequences of
incomplete Fibonacci and Lucas numbers, respectively. The same authors \cite{8} gave the following properties of the incomplete Fibonacci
and Lucas $p$-numbers:
\begin{equation}
\sum\limits_{j=0}^{h}\binom{h}{j}F_{p}^{k+j}\left( n+p\left( j-1\right)
\right) =F_{p}^{k+h}\left( n+\left( p+1\right) h-p\right)  \label{8}
\end{equation}
for $0\leq k\leq \frac{n-p-h-1}{p+1}$,
\begin{equation}
\sum\limits_{j=0}^{h}\binom{h}{j}L_{p}^{k+j}\left( n+p\left( j-1\right)
\right) =L_{p}^{k+h}\left( n+\left( p+1\right) h-p\right)  \label{9}
\end{equation}%
for $0\leq k\leq \frac{n-p-h}{p+1}.$

In this paper, we give the generalization of some results of \cite{3}. Some properties for the\ Fibonacci and Lucas $p$-numbers are obtained via the symmetric method. The results of incomplete Fibonacci and Lucas $p$-numbers are given by using binomial inverse pair as used for the Euler-Seidel matrix \cite{2,3}.

\section{Applications of symmetric infinite matrix}

\subsection{Applications for the Fibonacci and Lucas $p$-numbers}

Let us consider the initial sequences $a_{n}^{0}=F_{p}\left( n-1\right)$
and $a_{0}^{n}=F_{p}\left( n\left( p+1\right) -1\right) ,\ n\geq 1.$ Thus
the following infinite matrix is given for the special case

\begin{equation*}
\left(
\begin{array}{ccccc}
0 &amp; F_{p}\left( 0\right) &amp; F_{p}\left( 1\right) &amp; F_{p}\left( 2\right) &amp;
\cdots \\
F_{p}\left( p\right) &amp; F_{p}\left( p+1\right) &amp; F_{p}\left( p+2\right) &amp;
F_{p}\left( p+3\right) &amp; \cdots \\
F_{p}\left( 2p+1\right) &amp; F_{p}\left( 2p+2\right) &amp; F_{p}\left( 2p+3\right)
&amp; F_{p}\left( 2p+4\right) &amp; \cdots \\
F_{p}\left( 3p+2\right) &amp; F_{p}\left( 3p+3\right) &amp; F_{p}\left( 3p+4\right)
&amp; F_{p}\left( 3p+5\right) &amp; \cdots \\
\vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots%
\end{array}
\right) .
\end{equation*}

\begin{proposition}
The Fibonacci $p$-numbers satisfy the relation%
    \begin{equation}
        \sum\limits_{i=1}^{n}F_{p}\left( i\left( p+1\right) -1\right) =F_{p}\left(
        n\left( p+1\right) \right) .  \label{10}
    \end{equation}
\end{proposition}

\begin{proof}
For $a_{n}^{0}=F_{p}\left( n-1\right) $ and $a_{0}^{n}=F_{p}\left( n\left(
p+1\right) -1\right) ,\ n\geq 1$. We have $a_{1}^{1}=F_{p}\left( p+1\right)
, \  a_{1}^{2}=F_{p}\left( 2p+2\right) .$ Suppose that the equation holds for $
n&gt;1.$ Now we show that the equation holds for $\left( n+1\right) .$ Thus we get
using (\ref{4}) and (\ref{6})
    \begin{eqnarray*}
        a_{1}^{n+1} &amp;=&amp;a_{0}^{n+1}+a_{1}^{n} \\
        &amp;=&amp;F_{p}\left( \left( n+1\right) \left( p+1\right) -1\right) +F_{p}\left(
        n\left( p+1\right) \right) \\
        &amp;=&amp;F_{p}\left( n\left( p+1\right) +p\right) +F_{p}\left( n\left( p+1\right)
        \right) \\
        &amp;=&amp;F_{p}\left( n\left( p+1\right) +p+1\right) \\
        &amp;=&amp;F_{p}\left( \left( n+1\right) \left( p+1\right) \right) .
    \end{eqnarray*}%
By considering (\ref{5}), we have
    \begin{eqnarray*}
        a_{1}^{n} &amp;=&amp;\sum\limits_{i=1}^{n}\binom{n-i}{0}a_{0}^{i}+\sum
        \limits_{j=1}^{1}\binom{n-j}{n-1}a_{j}^{0} \\
        &amp;=&amp;\sum\limits_{i=1}^{n}F_{p}\left( i\left( p+1\right) -1\right) +a_{1}^{0}\\
        &amp;=&amp;\sum\limits_{i=1}^{n}F_{p}\left( i\left( p+1\right) -1\right)
        +F_{p}\left( 0\right) .
    \end{eqnarray*}
Then, we can obtain
    \begin{equation*}
        F_{p}\left( n\left( p+1\right) \right) =\sum\limits_{i=1}^{n}F_{p}\left(
        i\left( p+1\right) -1\right) .
    \end{equation*}
\end{proof}

Taking $p=1$ in (\ref{10}), we get $F_{2n}=\sum\limits_{i=1}^{n}F_{2i-1}$ in
\cite{4}.

Stakhov and Rozin \cite{7} gave the equation $F_{p}\left( 1\right) +F_{p}\left( 2\right) +\cdots +F_{p}\left( n\right)
=F_{p}\left( n+p+1\right) -1$ for the Fibonacci $p$-numbers. The following proposition shows that the formula can be
obtained via the symmetric method.

\begin{proposition}
The Fibonacci $p$-numbers are
    \begin{equation}
        \sum\limits_{j=1}^{n}F_{p}\left( j-1\right) =F_{p}\left( p+n\right) -1.
        \label{11}
    \end{equation}
\end{proposition}

\begin{proof}
Let $a_{n}^{0}=F_{p}\left( n-1\right) $ and $a_{0}^{n}=F_{p}\left( n\left(
p+1\right) -1\right) ,\ n\geq 1.$ If we take $n=1$ and $n=2$, then $
a_{1}^{1}=F_{p}\left( p+1\right) ,$ $a_{2}^{1}=F_{p}\left( p+2\right) .$
Suppose that the equation holds for $n&gt;1.$ We show that the equation holds for $
\left( n+1\right) $. We have by (\ref{4}) and (\ref{6})
    \begin{eqnarray*}
        a_{n+1}^{1} &amp;=&amp;a_{n}^{1}+a_{n+1}^{0} \\
        &amp;=&amp;F_{p}\left( p+n\right) +F_{p}\left( n\right) \\
        &amp;=&amp;F_{p}\left( p+n+1\right) .
    \end{eqnarray*}
Using (\ref{5}), we can write
    \begin{eqnarray*}
        a_{n}^{1} &amp;=&amp;\sum\limits_{i=1}^{1}\binom{n-i}{n-1}a_{0}^{i}+\sum
        \limits_{j=1}^{n}\binom{n-j}{0}a_{j}^{0} \\
        &amp;=&amp;a_{0}^{1}+\sum\limits_{j=1}^{n}a_{j}^{0} \\
        &amp;=&amp;F_{p}\left( p\right) +\sum\limits_{j=1}^{n}F_{p}\left( j-1\right),
    \end{eqnarray*}
which completes the proof.
\end{proof}

When $p=1$ in (\ref{11}), we obtain $\sum\limits_{i=1}^{n}F_{i}=F_{n+2}-1$
in \cite{4}.

In particular, let $a_{n}^{0}=L_{p}\left( n-1\right) $ and $a_{0}^{n}=L_{p}
\left( n\left( p+1\right) -1\right) ,\ n\geq 1.$ This case gives the
following infinite matrix
\begin{equation*}
\left(
\begin{array}{ccccc}
0 &amp; L_{p}\left( 0\right) &amp; L_{p}\left( 1\right) &amp; L_{p}\left( 2\right) &amp;
\cdots \\
L_{p}\left( p\right) &amp; L_{p}\left( p+1\right) &amp; L_{p}\left( p+2\right) &amp;
L_{p}\left( p+3\right) &amp; \cdots \\
L_{p}\left( 2p+1\right) &amp; L_{p}\left( 2p+2\right) &amp; L_{p}\left( 2p+3\right)
&amp; L_{p}\left( 2p+4\right) &amp; \cdots \\
L_{p}\left( 3p+2\right) &amp; L_{p}\left( 3p+3\right) &amp; L_{p}\left( 3p+4\right)
&amp; L_{p}\left( 3p+5\right) &amp; \cdots \\
\vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots%
\end{array}
\right) .
\end{equation*}

Similar results for the Lucas $p$-numbers can be obtained likewise. Therefore we omit the proofs of Proposition $4$ and $5$

\begin{proposition}
The Lucas $p$-numbers $L_{p}\left( n\right) $ satisfy the following relation%
    \begin{equation}
        \sum\limits_{i=1}^{n}L_{p}\left( i\left( p+1\right) -1\right) =L_{p}\left(
        n\left( p+1\right) \right) -\left( p+1\right) .  \label{12}
    \end{equation}
\end{proposition}

\begin{proposition}
We have%
    \begin{equation}
        \sum\limits_{j=1}^{n}L_{p}\left( j-1\right) =L_{p}\left( p+n\right) -1.
        \label{13}
    \end{equation}
\end{proposition}

If $p=1$ in (\ref{12}) and (\ref{13}), we get the well known identities $\sum\limits_{i=1}^{n}L_{2i-1}=L_{2n}-2\ $ and $\sum\limits_{i=1}^{n}L_{i}=L_{n+2}-3$.

\subsection{Applications for the incomplete Fibonacci and Lucas $p$-numbers}

In this subsection, we get similar formulas for (\ref{8}) and (\ref{9}) by
using the binomial inverse pair.\

Let $a_{n}^{0}=F_{p}^{k+n}\left( t+p\left( n-1\right) \right)$. From (\ref%
{2}) we have
\begin{equation*}
a_{0}^{n}=\sum\limits_{j=0}^{n}\binom{n}{j}F_{p}^{k+j}\left( t+p\left(
j-1\right) \right) .
\end{equation*}%
By using (\ref{8})
\begin{equation*}
a_{0}^{n}=F_{p}^{k+n}\left( t+\left( p+1\right) n-p\right) .
\end{equation*}%
Therefore, the dual formula of (\ref{8}) is obtained from (\ref{3})
\begin{equation}
F_{p}^{k+n}\left( t+p\left( n-1\right) \right) =\sum\limits_{j=0}^{n} \binom{%
n}{j}\left( -1\right) ^{n-j}F_{p}^{k+j}\left( t+\left( p+1\right) j-p\right)
\label{14}
\end{equation}
for $0\leq k\leq \frac{t-p-n-1}{p+1}.$ Similarly, let us take $%
a_{n}^{0}=L_{p}^{k+n}\left( t+p\left( n-1\right) \right) .$  Then (\ref{2}) can be rewritten as

\begin{equation*}
a_{0}^{n}=\sum\limits_{j=0}^{n}\binom{n}{j}L_{p}^{k+j}\left( t+p\left(
j-1\right) \right) .
\end{equation*}
By (\ref{9})
\begin{equation*}
a_{0}^{n}=L_{p}^{k+n}\left( t+\left( p+1\right) n-p\right) .
\end{equation*}
Finally, using (\ref{3}), we obtain the dual formula (\ref{9})
\begin{equation}
L_{p}^{k+n}\left( t+p\left( n-1\right) \right) =\sum\limits_{j=0}^{n} \binom{%
n}{j}\left( -1\right) ^{n-j}L_{p}^{k+j}\left( t+\left( p+1\right) j-p\right)
\label{15}
\end{equation}
for $0\leq k\leq \frac{t-p-n}{p+1}.$

For $p=1$ in (\ref{14}) and (\ref{15}), we get the properties of incomplete
Fibonacci and Lucas numbers in \cite{3}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% bibliography %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{9}
\bibitem{1} M.~Basu and B.~Prasad, \newblock 
Coding theory on the $m$-extension of the Fibonacci $p$-numbers,
\newblock {\em Chaos Solitons Fractals} {\bf 42} (2009), 2522--2530.

\bibitem{2} D.~Dumont, \newblock Matrices d'Euler-Seidel,
\newblock {\em S\'eminaire
Lotharingien de Combinatoire}, 1981, B05c.  Available
electronically at \newline
\url{http://www.emis.de/journals/SLC/opapers/s05dumont.pdf}.

\bibitem{3} A.~Dil and I.~Mez\H{o}, \newblock A symmetric algorithm for
hyperharmonic and Fibonacci numbers,
\newblock {\em Appl. Math. Comput.} {\bf 206} (2008), 942--951.

\bibitem{4} T.~Koshy,
\newblock {\em Fibonacci and Lucas Numbers with
Applications}, \newblock Wiley-Interscience, 2001.

\bibitem{5} E. G.~Kocer, N.~Tuglu, and A.~Stakhov, \newblock On the 
$m$-extension of the Fibonacci and Lucas $p$-numbers, \newblock {\em Chaos
Solitons Fractals} {\bf 40} (2009), 1890--1906.

\bibitem{7} A.~Stakhov and B.~Rozin, \newblock Theory of Binet formulas for
Fibonacci and Lucas $p$-numbers,
\newblock {\em Chaos Solitons Fractals} {\bf 27} (2006), 1162--1177.

\bibitem{8} D.~Tasci and M.~Cetin-Firengiz, \newblock Incomplete Fibonacci
and Lucas $p$-numbers,
\newblock {\em Math. Comput. Model.}
{\bf 52} (2010), 1763--1770.

\bibitem{9} N.~Tuglu, E.~G.~Kocer, and A.~Stakhov, \newblock Bivariate
Fibonacci like $p$-polynomials,
\newblock {\em Appl. Math. Comput.} {\bf 217} (2011), 10239--10246.

\bibitem{10} J. Riordan, \newblock {\em Combinatorial Identities},
Wiley, 1968.

\end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip \hrule
\bigskip

\noindent 2010 \textit{Mathematics Subject Classification}: Primary 11B73;
Secondary 11B83.

\noindent \emph{Keywords:} Symmetric infinite matrix, Fibonacci
$p$-number, Lucas $p$-number, incomplete Fibonacci $p$-number.

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

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received  August 16 2013;
revised versions received November 17 2013;
December 17 2013.
Published in {\it Journal of Integer Sequences},
January 6 2014.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in


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