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\vskip 1cm
{\LARGE\bf Integral Representations of Catalan and Related Numbers} \\
\vskip 1.5cm
\large K.A.Penson \\
\large J.-M. Sixdeniers \medskip \\
Universit\'{e} Pierre et Marie Curie \\
Laboratoire   de  Physique   Th\'{e}orique  des  Liquides \\
Tour 16, $5^{i\grave{e}me}$ \'{e}tage, 4, place Jussieu, 75252 Paris Cedex 05, France \\
\medskip
Email addresses:
\href{mailto:penson@lptl.jussieu.fr}{penson@lptl.jussieu.fr} and
\href{sixdeniers@lptl.jussieu.fr}{sixdeniers@lptl.jussieu.fr}

\vskip2.5cm
{\bf Abstract}
\end{center}

{\em
We   derive integral representations  for the  Catalan numbers $C(n)$,
shifted Catalan numbers $C(n+p)$, and the numbers $n!\cdot
C(n)$   and $C(n)\cdot  B(n)$, where  $B(n)$  are the  Bell numbers, for
$n=0,1\ldots$.  Our method is to use
inverse  Mellin transform. All these
numbers are power     moments   of positive functions,    and    their
representations turn out to be unique.
}

\bigskip

The Catalan numbers $C(n)$, $n=0,1,2,\ldots$, defined by
\begin{equation}
C(n)=\frac{\scriptsize{\left(\!\!\begin{array}{c}        2n         \\
n\end{array}\!\!\right)}}{n+1} \hspace{1cm},
\end{equation}
are among   the   most ubiquitous   sequences  in  enumerative
combinatorics.  Stanley \cite{Stanley} cites no less than 66 different
combinatorial settings where these  numbers appear. The first few  Catalan
numbers are
$$1,1,2,5,14,42,132,429,1430,4862$$
for $n=0\ldots9$. A   plethora  of
information about the $C(n)$'s can be found in \cite{Sloane}, under
sequence no. \htmladdnormallink{A000108}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000108}.

In this note we derive an integral representation of $C(n)$ as the $n$-th
power moment of a certain non-negative function $W_C(x)$ on the positive
half-axis. We also study the  ramifications of this representation for other integer
sequences involving $C(n)$.

To this end we seek a function $W_C(x)$ such that 
\begin{eqnarray}
\int_0^{\infty}x^nW_C(x)dx      &    =       &    C(n)\\  
&  =  &\frac{4^n\Gamma(n+1/2)}{\sqrt{\pi}\Gamma(n+2)}\hspace{1cm},\hspace{1cm}
n=0,1,\ldots\hspace{0.5cm}. \label{gam}
\end{eqnarray}
Replacing $n$ by a complex variable $s-1$, we rewrite Eq.(\ref{gam}) as 
\begin{equation}
\int_0^{\infty}x^{s-1}W_C(x)dx
=\frac{4^{s-1}\Gamma(s-1/2)}{\sqrt{\pi}\Gamma(s+1)}\hspace{0.5cm},\hspace{0.5cm}\mbox{\rm
Re}\hspace{0.2cm} s>1\hspace{0.5cm},\label{gam1}  \end{equation} which
implies that 
\begin{equation}
W_C(x)={\cal
M}^{-1}\left[\frac{4^{s-1}\Gamma(s-1/2)}{\sqrt{\pi}\Gamma(s+1)};x\right],\label{gam3}
\end{equation}
where ${\cal  M}^{-1}\left[f^*(s);x\right]=f(x)$ is the inverse Mellin
transform  \cite{Sneddon},         with     $f^*(s)={\cal
M}\left[f(x);s\right]=\int_0^{\infty}x^{s-1}f(x)dx\hspace{0.2cm}$       the
Mellin transform of $f(x)$.  We note the  following property of ${\cal
M}$ \cite{Sneddon} :
\begin{equation}  
{\cal
M}\left[x^bf(ax^h);s\right]=\frac{1}{h}a^{-\frac{s+b}{h}}f^*\left(\frac{s+b}{h}\right),\hspace{2cm}b\in
R ,\hspace{0.5cm}h>0,\label{Mellin1}
\end{equation}
which, when specialized  to $a=\frac{1}{4}$, $b=-\frac{1}{2}$ and $h=1$,
implies that 
\begin{equation}
{\cal
M}\left[x^{-\frac{1}{2}}f\left(\frac{x}{4}\right);s\right]=4^sf^*(s-1/2)/2\hspace{0.5cm}.
\label{Mellin2}
\end{equation}
Adopting  the     standard notation $(y)^\alpha_+=y^\alpha$  if  $y>0$, 
$(y)^\alpha_+=0$   otherwise,  and  using the formula 2.2(1), p.151 of
\cite{Marichev} :
\begin{equation}
{\cal M}\left[(1-x)^{\alpha-1}_+;s\right]=\Gamma(\alpha)\frac{\Gamma(s)}{\Gamma(\alpha+s)}
\hspace{0.5cm} ,\hspace{2cm}\alpha>0,\hspace{0.3cm} s>0,\label{Mellin3}
\end{equation}
we    can  apply    Eq.(\ref{Mellin2})   with    $f(x)=(1-x)^{\alpha-1}_+$  and
$\alpha=\frac{3}{2}$. This yields 
\begin{equation}
W_C(x)=\frac{x^{-\frac{1}{2}}}{\pi}\left(1-\frac{x}{4}\right)^{\frac{1}{2}}_+.\label{Mellin0}
\end{equation}
The   function $W_C(x)$  is   displayed on   Fig.(\ref{fig1}).  The  desired
integral representation of $C(n)$ is then
\begin{equation}
C(n)=\int_0^{4}x^n\left(\frac{\sqrt{\frac{4-x}{x}}}{2\pi}\right)dx
\hspace{0.5cm}. \label{catal1}
\end{equation}
This is the solution of the Hausdorff  moment problem on $[0,4]$, which
is  always  unique   \cite{Akhiezer}, and so   the  representation  of
Eq.(\ref{catal1}) is also unique.
\begin{figure}[htb]
\begin{center}
\epsfxsize=2in
\leavevmode\epsffile{FIG1CAT.eps}
\end{center}
\caption{\label{fig1} : The function $W_C(x)$, s. Eq.(9). This function diverges at $x=0$.}
\end{figure}

By the same token we can find the solution of 
\begin{equation}
\int_0^{\infty}x^nW_{C,p}(x)dx=C(n+p),\hspace{0.5cm} n=0,1,2\ldots,\hspace{0.5cm}p=1,2,\ldots,\label{catalan2}
\end{equation}
i.e. the unique representation  of the shifted Catalan numbers $C(n+p)$, as
the Hausdorff moments of 
\begin{equation}
W_{C,p}(x)=\frac{x^{p-\frac{1}{2}}}{\pi}\left(1-\frac{x}{4}\right)^{\frac{1}{2}}_+.
\end{equation}
The Mellin convolution property  for products of Mellin transforms, in
its simplest incarnation, states (\cite{Sneddon}, \cite{Marichev}) that
if $ {\cal M}\left[W_{1,2}(x);s\right]=\rho_{1,2}(s)$ then 
\begin{equation}
{\cal                    M}^{-1}\left[\rho_1(s)\rho_2(s);x\right]=W_{12}(x)\equiv
\int_0^{\infty}\frac{1}{t}W_1\left(\frac{x}{t}\right)W_2(t)dt\hspace{0.5cm}.\label{conv1}
\end{equation}
Observe that $ W_{1,2}(x)>0$ implies $ W_{12}(x)>0$.

As  an  application   of Eq.(\ref{conv1})   we look for   an integral
representation   of the sequence  $n!\cdot  C(n)$ whose initial terms are
$1,1,4,30,336,5040,95040,2162160,57657600,1764322560$, for
$n=0\ldots9$; compare \cite{Sloane}, no. \htmladdnormallink{A001761}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001761}.
 Using Eq.(\ref{Mellin0})   and
performing  the  Mellin    convolution  in Eq.(\ref{conv1}) with
$W_1(x)=e^{-x}$ and $W_2(x)=W_C(x)$, one  ends up with the following Stieltjes moment problem :

\begin{equation}
\int_0^{\infty}x^nW_{1C}(x)dx=n!\cdot
C(n)=\frac{(2n)!}{(n+1)!}\hspace{0.5cm},\hspace{0.5cm}n=0,1,\ldots,\label{weight1}
\end{equation}
with the solution 
\begin{eqnarray}
W_{1C}(x)                         &                    =             &
\frac{1}{2\pi\sqrt{x}}\int_{\frac{x}{4}}^{\infty}e^{-t}\frac{\sqrt{4t-x}}{t}dt\\
&              =            &            -\frac{1}{2}+\frac{1}{\sqrt{\pi
x}}e^{-\frac{x}{4}}+\frac{1}{2}\mbox{\rm
erf}\left(\frac{\sqrt{x}}{2}\right)\hspace{0.5cm},\label{16}
\end{eqnarray}
where erf$(y)$ is  the error  function. The  function $  W_{1C}(x)$ is
shown in Fig.(\ref{fig2}).  As  $W_{1C}(x)>0$,  the  (sufficient) Carleman
condition ($\sum_{n=1}^{\infty}(\frac{(2n)!}{(n+1)!})^{-\frac{1}{2n}}=\infty)$ (cf.
Ref.\cite{Akhiezer})  indicates  that the  solution $W_{1C}(x)$ of
Eq.(\ref{16}) is also unique.  Similar results are obtained by using 
$ W_{C,p}(x)$ instead of $W_C(x)$ in Eq.(\ref{conv1}).
\begin{figure}[htb]
\begin{center}
\epsfxsize=2in
\leavevmode\epsffile{FIG2CAT.eps}
\end{center}
\caption{\label{fig2} : The function $W_{1C}(x)$, s. Eq.(16). This function diverges at $x=0$.}
\end{figure}

Another use   of Eq.(\ref{conv1}) is   illustrated by  considering the
sequence $ C(n)\cdot B(n)$, where $B(n)$ are the Bell numbers ( see 
\cite{Sloane},  no. \htmladdnormallink{A000110}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000110}, and     
\cite{Comtet} ). The initial terms of this sequence are   1,
1, 4, 25, 210, 2184, 26796, 376233, 5920200, 102816714,
for  $n=0\ldots9$. For this last sequence see \cite{Sloane}, no. \htmladdnormallink{A064299}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A064299}. 
 The  weight function  whose  $n$-th moment is equal  to
$B(n)$   is       \begin{equation}      W_B(x)=\frac{1}{e} \sum_{k=1}^{\infty
}\frac{\delta(x-k)}{k!}\hspace{1cm}\label{dob},
\end{equation}
which is  a  consequence of Dobi\'{n}ski  formula, $B(n)=\frac{1}{e}
\sum_{k=1}^{\infty }\frac{k^n}{k!}$, see
\cite{Comtet}. In Eq.(\ref{dob}), $\delta(y)$ is Dirac's delta function.
By Mellin convolution of $W_B(x)$  with $W_C(x)$ one
obtains 

\begin{equation}
W_{BC}(x)=\frac{1}{2\pi
e}\sum_{k=1}^{\infty}\frac{1}{kk!}\sqrt{\frac{4k-x}{x}}H\left(4-\frac{x}{k}\right)\hspace{1cm},\label{weightbc}
\end{equation}
which, via  Carleman's criterion, is the  only  positive function such
that    its   $n$-th   moment   is    equal  to    $C(n)\cdot  B(n)$.   In
Eq.(\ref{weightbc})  $ H(y)$ is the   Heaviside function. The function
$W_{BC}(x)$ is displayed on Fig.(\ref{fig3}).
\begin{figure}[htb]
\begin{center}
\epsfxsize=2in
\leavevmode\epsffile{FIG3CAT.eps}
\end{center}
\caption{\label{fig3} : The function $W_{BC}(x)$, s. Eq.(18). This function diverges at $x=0$.}
\end{figure}
\begin{figure}[htb]
\begin{center}
\epsfxsize=2in
\leavevmode\epsffile{FIG4CAT.eps}
\end{center}
\caption{\label{fig4} : The function $W_{3}(x)$, s. Eq.(24). This function diverges at $x=0$.}
\end{figure}

The last  sequence  that  will  concern  us  here   is
$(n!)^2C(n)=\frac{(2n)!}{n+1}$.  Its initial terms  are 
$$1,1,8,180,8064,604800,68428800,10897286400,2324754432000$$
for
$n=0\ldots9$; compare \cite{Sloane}, no. \htmladdnormallink{A060593}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A060593}.    Proceeding
as in Eqs.(3) and (4), we are looking for $W_3(x)$ satisfying 
\begin{equation}
\int_0^{\infty}x^{s-1}W_3(x)dx
=\frac{4^{s-1}\Gamma(s-1/2)\Gamma^2(s)}{\sqrt{\pi}\Gamma(s+1)}\hspace{1cm},\hspace{0.5cm}\mbox{\rm
Re}\hspace{0.2cm}s>1\hspace{0.5cm}.\label{W3}
\end{equation}
It appears that when studying Eq.(\ref{W3})  it is possible to avoid using
$W_C(x)$.  As the first  step we observe  from Eq.(6) that
\begin{equation}
{\cal                                                   M}^{-1}\left[\Gamma
\left(s-\frac{1}{2}\right);x\right]=\frac{e^{-x}}{\sqrt{x}}\hspace{1cm}. \label{W31}
\end{equation}
In addition, the following relation holds :
\begin{equation}
{\cal                M}^{-1}\left[\frac{\Gamma^2                     (s)}{\Gamma
(s+1)};x\right]=-Ei(-x)\hspace{1cm}, \label{W32}
\end{equation}
which  is   the    formula  8.1(1),   p.182    of  \cite{Marichev}. In
Eq.(\ref{W32})    $Ei(y)$  is   the  exponential  integral   function.
Combining Eqs.(\ref{W31}) and (\ref{W32}) in the Mellin convolution we
obtain 
\begin{eqnarray}
{\cal      M}^{-1}\left[\frac{\Gamma(s-1/2)\Gamma^2(s)}{\Gamma(s+1)};x\right]       &
=&-\frac{1}{\sqrt{x}}\int_0^{\infty}t^{-\frac{1}{2}}t^{-\frac{x}{t}}Ei(-t)dt
\label{W33}
\\ & = & 2
\sqrt{\frac{\pi}{x}}e^{-2\sqrt{x}}+ 4\sqrt{\pi}Ei(-2\sqrt{x})\hspace{1cm},\hspace{0.5cm}x>0\hspace{0.5cm}. \label{W34}
\end{eqnarray}
In writing  Eq.(\ref{W34}) we have  used the formula 2.5.4.2, p.72
of
\cite{Prudnik}. Finally, we use Eq.(6) again (with $a=\frac{1}{4}$,
$b=0$ and $h=1$) and from Eq.(19) we get the solution 
\begin{equation}
W_3(x)=\frac{1}{\sqrt{x}}e^{-\sqrt{x}}+Ei(-\sqrt{x})     \hspace{1cm},
\label{W35}
\end{equation}
which is plotted in  Fig.(\ref{fig4}). As $W_3(x)>0$,  by Carleman's criterion the
solution is again unique.

Remark:
E. P.Wigner \cite{Wig} has demonstrated
that Eq.(\ref{catal1}), under a suitable parametrization,
describes the distribution function of eigenvalues of an ensemble of random,
symmetric, real matrices.

Integral representations of other combinatorial  numbers can be  found
in \cite{Egorychev}.  For further  applications of  Mellin convolution
formula Eq.(\ref{conv1}), one  may consult \cite{Marichev}, \cite{ML},
\cite{SP1}, \cite{SP2} ,  \cite{KPS} and \cite{SPK}.

\begin{thebibliography}{99}

\bibitem{Akhiezer} N. I. Akhiezer, \emph{The Classical Moment Problem and Some Related Questions in Analysis}, (Oliver and Boyd, London, 1965)

\bibitem{Comtet} L. Comtet, \emph{Advanced Combinatorics}, (D. Reidel, Boston, 1984)

\bibitem{Egorychev} G. P. Egorychev, \emph{Integral Representation and
the Computation of  Combinatorial Sums}, Translations  of Mathematical
Monographs,   Vol. 59,  (American Mathematical   Society, Rhode Island,
1984)

\bibitem{KPS} J. R. Klauder, K. A. Penson and   J. M. Sixdeniers,
\emph{Constructing coherent states through solutions of Stieltjes and
Hausdorff moment problems}, Phys. Rev. \textbf{A64}, 013817 (2001)

\bibitem{Marichev} O. I. Marichev, \emph{Handbook of Integral
Transforms of Higher  Transcendental Functions, Theory and Algorithmic
Tables}, (Ellis Horwood Ltd, Chichester, 1983)

\bibitem{Prudnik} A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev,
\emph{Integrals and Series, vol. 2: Special Functions},  (Gordon and Breach, New York, 1998)

\bibitem{SP1}J. M. Sixdeniers and K. A. Penson, \emph{On the completeness of
coherent    states      generated    by     binomial    distribution},
J. Phys. \textbf{A33}, 2907 (2000)

\bibitem{SP2}J. M. Sixdeniers and K. A. Penson, \emph{On the completeness
of photon-added coherent states}, J. Phys. \textbf{A34}, 2859 (2001)

\bibitem{SPK} J. M. Sixdeniers, K. A. Penson and J. R. Klauder, \emph{Tricomi
coherent states}, Int. J. Mod. Phys. B  \textbf{15}, 4231 (2001)

\bibitem{ML}J. M. Sixdeniers, K. A. Penson and A. I. Solomon,
\emph{Mittag-Leffler coherent states}, J. Phys. \textbf{A32}, 7543 (1999)

\bibitem{Sloane}N. J. A.  Sloane, On-Line Encyclopedia of Integer
Sequences,             published           electronically          at:
http://www.research.att.com/$\sim$/njas/sequences/

\bibitem{Sneddon} I. N. Sneddon, \emph{The Use of Integral Transforms}, (McGraw-Hill, New York, 1974)

\bibitem{Stanley} R. P. Stanley, \emph{Enumerative Combinatorics}, Vol. 2, (Cambridge University Press, 1999)

\bibitem{Wig} E. P. Wigner, \emph{Characteristic vectors of bordered
matrices with infinite dimensions}, Ann.Math.\textbf{62}, 548 (1955)

\end{thebibliography}


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\vspace*{+.1in}
\noindent
{\small
(Concerned with sequences
\htmladdnormallink{A000108}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000108},
\htmladdnormallink{A000110}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000110},
\htmladdnormallink{A001761}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001761},
\htmladdnormallink{A060593}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A060593},
\htmladdnormallink{A064299}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A064299}.)
}

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\vspace*{+.1in}
\noindent
Received Sep 7, 2001; revised version received Oct 29, 2001.
Published in Journal of Integer Sequences, Feb 7, 2002.

\centerline{\rule{6.5in}{.01in}}

\vspace*{+.1in}
\noindent
Return to \htmladdnormallink{Journal of Integer Sequences home
page}{http://www.
research.att.com/~njas/sequences/JIS/}.

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