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\vskip 1cm{\LARGE\bf
A New Characterization  of Catalan Numbers \\
\vskip .03in
Related to Hankel Transforms and Fibonacci\\
\vskip .14in Numbers}

\vskip 1cm \large
Belgacem Bouras \\
Department of Mathematics \\
College of Sciences\\
Gab\`{e}s University\\
Tunisia\\
\href{mailto:Belgacem.Bouras@issatgb.rnu.tn}{\tt Belgacem.Bouras@issatgb.rnu.tn}\\
\href{mailto:Belgacem.Bouras.2003@voila.fr}{\tt Belgacem.Bouras.2003@voila.fr}\\
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\begin{abstract}
Cvetkovi\'c, Rajkovi\'c, and Ivkovi\'c  proved that the Hankel
transform of the sequence of sums of two successive Catalan numbers
is the sequence of Fibonacci numbers with odd indices. Later,
Benjamin, Cameron, Quinn, and Yerger  extended this result and
proved that if we remove one term from this sequence of sums, then the
Hankel transform is the sequence of Fibonacci numbers with even
indices. In this paper, we prove that the Catalan numbers are
the unique nonnegative integer sequence satisfying this property.
\end{abstract}


\section{Introduction}\label{sec:intro}

 The Hankel transform of a sequence $\{a_n\}_{n=0}^\infty$ is the sequence
 $H(\{a_n\}_{n=0}^\infty)=\{h_n\}_{n=0}^\infty$ defined as follows:
 \begin{equation}\label{eqn:1} h_n= \det[a_{i+j}]_{0 \leq i,\,j \leq n-1}.\end{equation}
 The Hankel transform was introduced and first studied by Layman \cite{L},
 who showed that the Hankel transform of a sequence is equal to the Hankel
 transform of the binomial  transform of this
 sequence and conjectured that   the Hankel transform of the sequence of sums of two adjacent Catalan numbers
 is a subsequence of Fibonacci number sequence.


Recall that the Catalan number sequence  $\{C_n\}_{n=0}^\infty$ is defined as follows:
  \[
C_n=\frac{{2n \choose n}}{n+1},\; n \geq0.
\]
This sequence is the  unique nonnegative integer sequence
$\{a_n\}_{n=0}^\infty$ fulfilling (see \cite{S})
\[\det[a_{i+j}]_{0
\leq i,\,j \leq n-1}=1\] and
\[\det[a_{i+j}]_{1
\leq i,\,j \leq n}=1.\] Cvetkovi\'c, Rajkovi\'c, and Ivkovi\'c
\cite{CRI}  proved the following formula, known as Layman's
conjecture,
\begin{equation}\label{eqn:5}
\det[C_{i+j+1}+C_{i+j+2}]_{0 \leq i,\,j \leq n-1}=F_{2n+2},\,n \geq
1.
 \end{equation}
 where $F_{n},\; n \geq 0$ is the $n$-th Fibonacci number
defined by the following recurrence:
  \begin{equation}\label{eqn:3} F_0=0,\,F_1=1,\; F_{n+2}=F_{n+1}+F_{n},\; n\geq 0.\end{equation}
 The proof depends on
special properties of the corresponding orthogonal polynomials.
Benjamin, Cameron, Quinn, and Yerger \cite{BCQY} used a
combinatorial approach and  proved that
\begin{equation}\label{eqn:2}
\det[C_{i+j}+C_{i+j+1}]_{0 \leq i,\,j \leq n-1}=F_{2n+1},\,n \geq 1.
 \end{equation}
In view of (\ref{eqn:2}) and (\ref{eqn:5}), it should be mentioned
here that (\ref{eqn:2}) and (\ref{eqn:5}) are special cases of the more
general determinant evaluation  given  in \cite{B2}.

A natural question arises: Is the Catalan number sequence   the
unique nonnegative integer sequence fulfilling  (\ref{eqn:2}) and
(\ref{eqn:5})? More precisely, is there any other nonnegative
integer sequence $\{a_n\}_{n=0}^\infty$ fulfilling
\begin{equation} \label{eqn:60}
\det[a_{i+j}+a_{i+j+1}]_{0 \leq i,\,j \leq n-1}=F_{2n+1},\,n \geq 1
 \end{equation}
 and
\begin{equation}\label{eqn:70}
\det[a_{i+j+1}+a_{i+j+2}]_{0 \leq i,\,j \leq n-1}=F_{2n+2},\,n \geq
1.
 \end{equation}
The aim of this paper is to answer this question. Namely, we prove the main theorem of this paper:
\begin{theorem}\label{thm:1}
The unique nonnegative integer sequence fulfilling (\ref{eqn:60})
and (\ref{eqn:70})
 is
the Catalan number sequence.
\end{theorem}
 Our paper is organized as follows:  In
Section \ref{sec:NPR}, we state some basic results concerning the
theory of orthogonal polynomial. These results will be used in
Section \ref{sec:proof} to prove the main theorem.
\section{Notation and preliminary results}\label{sec:NPR}
Let  $\mathcal{P}$ be the vector space of polynomials
with complex coefficients and let  $\mathcal{P}^{\prime }$ be its algebraic dual. We denote by $%
\left\langle u,f\right\rangle $ the action of $u\in
\mathcal{P}^{\prime }$ on $f$ $\in $ $\mathcal{P}.$ In particular, we
denote by $u _{n}=\left\langle
u,x^{n}\right\rangle,\,n\geq 0$ the moments of $u$.
 If $u\in \mathcal{P}^{\prime }$ and $\Phi \in $ $%
\mathcal{P}$, then the left multiplication of the functional \ $u$
by the
polynomial $\Phi ,$ denoted by $\Phi u,$ is the functional in $\mathcal{P}%
^{\prime }$ defined as follows:
\begin{equation}\label{eqn:8}\left\langle \Phi u,f\right\rangle
=\left\langle u,\Phi f\right\rangle,\,f\in
\mathcal{P}.\end{equation} \
\begin{definition} \label{defn:defn} (see \cite{C}) A sequence of polynomials
$\{P_{n}\}_{n\geq 0}$ is said to be a {\it monic orthogonal polynomial
sequence} (MOPS) with\ respect to a linear functional $u$ if

\noindent $i)$ $\deg P_{n}=n$ and the leading coefficient of
$P_{n}(x)$ is
equal to 1.\\
$ii)$ $\left\langle u,P_{n}P_{m}\right\rangle =r_{n}\delta
_{n,m},n,m\geq 0,r_{n}\neq 0,n\geq 0. $ \\
 A linear functional  $u$ is
 called \textit{regular} if there exists a polynomial sequence $\{P_{n}\}_{n\geq 0}$
  orthogonal with respect to $u$. Throughout this paper, we will take all regular linear
functionals $u$ normalized, i.e., $u_{0}=1.$ \end {definition}
According to Favard-Shohat theorem, a sequence of monic orthogonal
polynomials satisfies a three-term recurrence relation (see
\cite{C}):
\begin{equation}\label{eqn:9}
\begin{array}{c}
\ P_{0}(x)=1,\;P_{1}(x)=x-\beta _{0},\medskip \\
\ P_{n+2}=(x-\beta _{n+1})P_{n+1}(x)-\gamma _{n+1}P_{n}(x),n\geq 0,
\end{array}
\ \end{equation} with
 $$(\beta _{n},\gamma _{n})\in \mathbb{C} \times (\mathbb{C} \setminus
\left\{ 0\right\}) ,n\geq 0.$$
\section{Proof of the main theorem}\label{sec:proof}
First of all, recall that a linear functional $L$ is regular if and only if (see \cite{C})
\begin{equation}\label{eqn:10}
\Delta_n(L)\neq 0,
\end{equation}
where $\Delta_n(L)$ is the Hankel determinant of order $n+1$ of $L$
defined as follows:
\begin{equation}\label{eqn:30} \Delta _{-1}(L)=1,\; \Delta_{n}(L) =\left|
\begin{array}{cccc}
L_{0} &amp; L_{1} &amp; \cdots &amp; L_{n} \\
L_{1} &amp; L_{2} &amp; \cdots &amp; L_{n+1} \\
\vdots &amp; \vdots &amp;  &amp; \vdots \\
L_{n} &amp; L_{n+1} &amp; \cdots &amp; L_{2n}
\end{array}
\right| ,\, n\geq 0. \medskip \end{equation} In this condition, the
coefficient $\gamma _{n}$ in (\ref{eqn:9})\ is given as follows (see
\cite{B,C}):
\begin{equation}\label{eqn:11}\gamma _{n+1}=\frac{\Delta _{n+1}(L)\Delta _{n-1}(L)}{\Delta _{n}^{2}(L)},\,n\geq 0.\end{equation}
Conversely, the Hankel determinant $\Delta _{n}(L)$ can be written in terms of $\gamma
_{n}$ as follows (see \cite{B}):
\begin{equation}\label{eqn:12}\ \ \ \ \Delta _{-1}(L)=\Delta _{0}(L)=1,\ \Delta _{n}(L)=%
\prod_{k=1}^{n}\tau _{k},\,n\geq 0,\end{equation} where
\begin{equation}\label{eqn:13} \tau
_{k}=\prod_{i=1}^{k}\gamma _{i},\,k\geq 1. \end{equation} Let us
recall  a result established in  \cite{C1} concerning the so-called
{\it kernel polynomials}. Given a MOPS $\left\{ P_{n}\right\} _{n\geq 0}$
orthogonal with respect to a linear functional $L$. The linear
functional $L^\ast$ defined as follows:
$$L^\ast=\lambda(x-c)L,\; c \in \mathbb{C},$$
where $\lambda$ is a normalization factor, is regular if and only if
$$P_n(c) \neq 0,\; n\geq 0.$$
In this case the  MOPS corresponding to $L^\ast$, denoted by
$\left\{ P^*_{n}(c,x)\right\} _{n\geq 0}$, satisfies
$$P^*_{n}(c,x)=\frac{1}{x-c}(P_{n+1}(x)-\frac{P_{n+1}(c)}{P_{n}(c)}P_{n+1}(x)).$$
The sequence $\left\{ P^*_{n}(c,x)\right\} _{n\geq 0}$ is called the
sequence of kernel polynomials of K-parameter $c$ corresponding to
$\left\{ P_{n}\right\} _{n\geq 0}.$ The recurrence coefficients of
$\left\{ P^*_{n}(c,x)\right\} _{n\geq 0}$ denoted by $\beta^*_n$ and
$\gamma^*_{n+1}$ are expressed in terms of those of $\left\{
P_{n}\right\} _{n\geq 0}$ as follows:
\begin{equation}\label{eqn:14}\beta^*_n=\beta_{n+1}+\frac{P_{n+2}(c)}{P_{n+1}(c)}-
\frac{P_{n+1}(c)}{P_{n}(c)},\end{equation}
\begin{equation}\label{eqn:15} \gamma^*_{n+1}=\gamma_{n+1}\frac{P_{n+2}(c)P_{n}(c)}{P_{n+1}^2(c)},\,
n \geq 0.
\end{equation}
Let $\{a_n\}_{n=0}^\infty$ be a non-negative integer sequence
fulfilling (\ref{eqn:60}) and (\ref{eqn:70}).  Then for $n=1,$ we get
$a_0+a_1=F_{3}=2 \neq 0$ and $a_1+a_2=F_{4}=3 \neq 0.$ Define the  linear functionals $u,\;v$ and $v^*$ by
\begin{equation}\label{eqn:16}
u_n=\frac{C_n+C_{n+1}}{C_0+C_1},\; n\geq 0,
\end{equation}
\begin{equation}\label{eqn:17}
v_n=\frac{a_n+a_{n+1}}{a_0+a_1},\; n\geq 0,
 \end{equation}
 \begin{equation}\label{eqn:18}
v^*_n=\frac{a_{n+1}+a_{n+2}}{a_1+a_2},\; n\geq 0.
 \end{equation}
  From (\ref{eqn:30}), (\ref{eqn:16}) and (\ref{eqn:2}),   we have
 \begin{equation}\label{eqn:19}
 \Delta_n(u)=\frac{F_{2n+3} }{(C_0+C_1)^{n+1}},\; n \geq 0.
 \end{equation}
 Using the recurrence (\ref{eqn:3}),  we can easily see that $F_n \neq 0, \; n \geq 0.$
 Therefore,  the condition (\ref{eqn:10}) is satisfied by $u$. Hence, $u$ is regular.
 Similarly, from (\ref{eqn:30}), (\ref{eqn:60}), (\ref{eqn:70}), (\ref{eqn:17})
  and the relation (\ref{eqn:18}), we get
 \begin{equation}\label{eqn:20}
 \Delta_n(v)=\frac{F_{2n+3} }{(a_0+a_1)^{n+1}},\; n \geq 0
 \end{equation}
 and
 \begin{equation}\label{eqn:21}
 \Delta_n(v^*)=\frac{F_{2n+4} }{(a_1+a_2)^{n+1}},\; n \geq 0.
 \end{equation}
 For the same reason, $v $ and $v^*$ are regular. We will denote by $\left\{
P_{n}\right\} _{n\geq 0}$ and $\left\{ P^*_{n}(c,x)\right\} _{n\geq 0}$ the MOPS corresponding to $v $ and $v^*$ respectively.
  From (\ref{eqn:17})  and  (\ref{eqn:18}), we have
 $$v^*=\frac{a_0+a_1}{a_1+a_2}xv.$$
 So, the sequence $\left\{ P^*_{n}(c,x)\right\} _{n\geq 0}$ is the sequence of kernel polynomials of K-parameter $0$ corresponding to
$\left\{ P_{n}\right\} _{n\geq 0}.$ Hence, using (\ref{eqn:12}),
(\ref{eqn:13}) and (\ref{eqn:15}), with $c=0$, we get
 \[
 \Delta_{n}(v^*)=\frac{P_{n+1}(0)}{(P_{1}(0))^{n+1}}\Delta_{n}(v),
 \;n \geq0.
 \]
 Taking into account (\ref{eqn:20}) and (\ref{eqn:21}), we obtain
  \begin{equation}\label{eqn:23}
  F_{2n+4}=(\frac{a_1+a_2}{a_0+a_1})^{n+1}\frac{P_{n+1}(0)}{(P_{1}(0))^{n+1}}F_{2n+3},\; n\geq 0.
\end{equation}
Suppose that $\left\{P_{n}\right\} _{n\geq 0}$ satisfies the
recurrence (\ref{eqn:9}). We have
$$P_1(0)=-\beta_0.$$
But, from Definition \ref{defn:defn} and the relation (\ref{eqn:9}), we have
$$0=&lt;v,P_1P_0&gt;=&lt;v,P_1&gt;=&lt;v,x&gt;-\beta_0&lt;v,1&gt;=v_1-\beta_0.$$
Thus
$$\beta_0=v_1=\frac{a_1+a_2}{a_0+a_1}.$$
Therefore
\begin{equation}\label{eqn:24} P_1(0)=-\frac{a_1+a_2}{a_0+a_1} .\end{equation}
Substitution of (\ref{eqn:24}) in (\ref{eqn:23}) gives
$$
F_{2n+4}=(-1)^{n+1}P_{n+1}(0)F_{2n+3},\; n\geq 0 $$ or equivalently
\begin{equation}\label{eqn:25}
P_{n+1}(0) =(-1)^{n+1}\frac{F_{2n+4}}{F_{2n+3}},\; n\geq 0.
\end{equation}
From (\ref{eqn:11}) and (\ref{eqn:20}), we get
\begin{equation}\label{eqn:26}
\gamma_{n+1}=\frac{F_{2n+5}F_{2n+1}}{F_{2n+3}^2},\; n\geq 0.
\end{equation}
Taking $x=0$ in (\ref{eqn:9})  and using (\ref{eqn:25}) and (\ref{eqn:26}),  we get
$$
\beta_{n+1}=\frac{F_{2n+3}F_{2n+6}}{F_{2n+4}F_{2n+5}}+\frac{F_{2n+2}F_{2n+5}}{F_{2n+3}F_{2n+4}},\;
n\geq 0.
$$
On account of the recurrence relation (\ref{eqn:3}), we get
\begin{equation}\label{eqn:27}
\beta_{n+1}=\frac{F_{2n+4}}{F_{2n+3}}+\frac{F_{2n+3}}{F_{2n+5}},\;
n\geq 0.
\end{equation}
We have proved that the recurrence coefficients of the MOPS
corresponding to  any sequence $\{a_n\}_{n=0}^\infty$ satisfying (\ref{eqn:60}) and (\ref{eqn:70})
 are given by (\ref{eqn:26}) and (\ref{eqn:27}). Since the sequence $\{C_n\}_{n=0}^\infty$
itself satisfies (\ref{eqn:60}) and (\ref{eqn:70})   then the recurrence coefficients of its MOPS
are also given by  (\ref{eqn:26}) and (\ref{eqn:27}). Consequently, by the uniqueness of the linear
functional corresponding to a MOPS, we have
$$u=v$$
or, equivalently,
$$u_n=v_n,\; n \geq0.$$
Therefore, owing to (\ref{eqn:16}) and (\ref{eqn:17}), we obtain
\begin{equation}\label{eqn:28}
a_n+a_{n+1}=C_n+C_{n+1},\; n\geq0.
\end{equation}
On the other hand, we have
$$a_0+a_{1}=F_3=2.$$
 Taking into account  the fact that
$\{a_n\}_{n=0}^\infty$ is a nonnegative sequence, we get $a_0=1=C_0$
and
$a_1=1=C_1.$
By induction and using (\ref{eqn:28}), we can easily prove that,
$$a_n=C_n,\; n \geq0.$$
Which completes the proof.
\section{Acknowledgments} We are grateful to the anonymous referee for careful reading
the text and essential remarks.

\begin{thebibliography}{99}




\bibitem{BCQY} A.~T.~Benjamin, N.~T.~Cameron, J.~J.~Quinn, and C.~R.~Yerger,
Catalan determinants --- a combinatorial approach, {\it Congr.
Numer.} {\bf 200} (2010), 27--34.
\bibitem{B}{B. Bouras, Generalized Jacobi orthogonal polynomials. {\it Integral Transf. and Special Functions}, {\bf 18}
(2007),  715--730. }
\bibitem{B2}{B. Bouras, Hankel determinant of a linear combination of three successive Catalan numbers.
{\it Mediterr. J. Math}.
(In press). }
\bibitem{C1}{T. S. Chihara, On co-recursive  orthogonal polynomials, {\it Proc. Amer. Math. Soc.}, {\bf 8} (1957), 899--905.}

\bibitem{C}{T. S. Chihara, {\it An Introduction to Orthogonal Polynomials}, Gordon and Breach, New York, (1978)}

\bibitem{CRI} A. Cvetkovi\'c, P. Rajkovi\'c, and
M. Ivkovi\'c, Catalan numbers, and Hankel transform, and Fibonacci
numbers, {\it J. Integer Seq.} {\bf 5} (2002),
\href{http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Ivkovic/ivkovic3.html}{Article
02.1.3}.

\bibitem{L}
J.~W. Layman,
\href{http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html}
{The {H}ankel transform and some of its properties}, \emph{J.
  Integer Seq.} \textbf{4} (2001), Article 01.1.5.
\bibitem{S}
R.~P. Stanley, \emph{Enumerative Combinatorics}, Vol.~1, Cambridge
University
  Press, 1997.


\bibitem{OEIS} N.~J.~A.~Sloane, {\it The On-Line Encyclopedia of Integer Sequences.} Published electronically at \href{http://oeis.org}{\tt http://oeis.org}.

\end{thebibliography}


\bigskip
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\noindent {\it 2010 Mathematics Subject Classification:} Primary: 33C45
; Secondary  42C05.

\noindent {\it Keywords:} Hankel transform, Catalan number,
recurrence relation, integer sequence.


\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences \seqnum{A000159},
\seqnum{A014523}, and \seqnum{A103433}.)
\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received July 11 2012;
revised versions received  October 23 2012; December 28 2012; 
February 3 2013.
Published in {\it Journal of Integer Sequences}, March 2 2013.

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