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\begin{center}
\vskip 1cm{\LARGE\bf Brownian Motion and the}\\
\vskip 0.1in{\LARGE\bf Generalized Catalan Numbers}
\vskip 1cm \large
Joseph Abate\\
900 Hammond Road\\
Ridgewood, NJ 07450-2908\\
USA\\
\ \\
Ward Whitt\\
Department of Industrial Engineering and Operations Research\\
Columbia University\\
New York, NY 10027-6699\\
USA\\
\href{mailto:ww2040@columbia.edu}{\tt ww2040@columbia.edu} \\
\end{center}

\vskip .2 in

\begin{abstract}
We show that the generating functions of the generalized Catalan numbers can be identified with the moment generating functions
of probability density functions related to the Brownian motion stochastic process.
Specifically, the probability density functions are exponential mixtures of inverse Gaussian (EMIG) probability density functions,
which arise as the first passage time distributions to the origin of Brownian motion with a negative drift and an exponential initial distribution on the positive halfline.
As a consequence of the EMIG representation, we show that the generalized Catalan numbers are the moments of generalized beta distributions.
We also study associated convolution sequences arising as the coefficients of the product of two generalized Catalan generating functions.
\end{abstract}

\providecommand{\e}[1]{\ensuremath{\times 10^{#1}}}

\section{Introduction}\label{secIntro}

Our purpose in this paper, as in our recent \cite{AW10}, is to
establish connections between probability theory and integer
sequences.  We show how established probability results can be
applied to generate new integer sequences and results about
integer sequences, after appropriate connections have been
established.
In particular, we establish a connection between the classical
Brownian motion stochastic process and the generalized Catalan
numbers, $C_n (\alpha)$. We show that the generalized Catalan
numbers are intimately connected to certain exponential
mixtures of inverse Gaussian distributions, which arise as
first passage times to the origin of Brownian motion with
negative drift, starting with an exponential initial
distribution on the positive halfline.  (See Theorems \ref{th1}
and \ref{thBetaMoments}.)  We then apply this relation to identify
interesting integer sequences and relations among integer
sequences.

As a consequence of our analysis, we propose the new family of
integer sequences $\{V_n (\alpha): n \ge 1\}$, where $\alpha$
is a positive integer and
\beql{int1}
V_n (\alpha) \equiv \sum_{k = 0}^{n} \binom{n+k}{k} \alpha^k, \quad n \ge 1,
\eeq
where $\equiv$ denotes ``equality by definition.''
The integer sequence $\{V_n (1)\}$ (\seqnum{A001700}) is pervasive in the OEIS \cite{S10}, but we ourselves only recently contributed
the integer sequence $\{V_n (2)\}$ (\seqnum{A178792}).  A primary goal is to expose the structure of the family $\{V_n (\alpha)\}$, $\alpha \ge 1$.
The general sequence $\{V_n (\alpha)\}$ has been studied on p. 236 of \cite{GKP94}; see pp. 167, 215 for more on the special case $\alpha = 1/2$.

In the end, there is a fairly direct connection between the classical sequence of Catalan numbers $\{C_n\}$, with
\beql{int1a}
C_n \equiv \binom{2n}{n}\frac{1}{n+1}  = \frac{(2n)!}{n!(n+1)!}, \quad n \ge 1,
\eeq
 and the integer sequences $\{V_n (\alpha)\}$
via their generating functions, which we summarize now:
\begin{eqnarray}
c(x) &amp; \equiv &amp; \sum_{n=0}^{\infty} C_n x^n = \frac{2}{1 + \sqrt{1-4x}}, \label{a1} \\
c(x; \alpha) &amp; \equiv &amp; \sum_{n=0}^{\infty} C_n (\alpha) x^n = (1 - x c(\alpha x))^{-1}, \label{a1a} \\
c(x; a,b) &amp; \equiv &amp; \sum_{n=0}^{\infty} C_n (a,b) x^n \equiv c(bx; a) c(ax; b), \label{c1} \\
v(x; \alpha) &amp; \equiv &amp; \sum_{n=0}^{\infty} V_n (\alpha) x^n \equiv c(2 \alpha x; 1/2) c(x; \alpha); \label{d2}.
\end{eqnarray}
%see \eqn{a1}, \eqn{a1a}, \eqn{c1} and \eqn{d2} below.
Supporting theory appears in Theorems \ref{th2}-\ref{thVcf}.
In Theorem \ref{thBetaMoments} and \ref{thIntRepV} we establish integral representations, which are known to provide additional insight \cite{P02}.
Many examples of integral representations appear in \cite{S10}.

The relations in \eqn{a1a}--\eqn{d2} are especially interesting to us, because they are generalizations of important relations for
functions characterizing the transient behavior of reflected Brownian motion exposed in our first paper \cite{AW87}.
Since the transient mean is nondecreasing and bounded, it can be regarded as a probability cumulative distribution function (cdf)
when we divide by the limiting value; in \cite{AW87} we called it the ``RBM first-moment cdf.''
   We discuss the connections between relations \eqn{a1}-\eqn{d2} and our previous papers \cite{AW87,AW96,AW99c} in \S \ref{secPapers}.

\section{Background}

We start by giving background on the generalized Catalan numbers, Brownian motion and associated first passage time distributions.



\paragraph{The generalized Catalan numbers.}
The Catalan numbers in \eqn{int1a} frequently arise in combinatorics and are pervasive in the OEIS \cite{S10}; see \seqnum{A000108}.
Following Lang (\seqnum{A064062}), we define the {\em generalized Catalan numbers} $C_n (\alpha)$ as the coefficients of the generating function
$c(x; \alpha)$ defined in \eqn{a1a},
where $c(x)$ is the generating function of the (ordinary) Catalan numbers $C_n$ in \eqn{a1}.
Other proposed definitions for (properties of) the generalized Catalan numbers appear in
\seqnum{A006633}, \seqnum{A068765}, \seqnum{A130564} and in \cite[p.\ 14]{B06}.
We prefer definition \eqn{a1} because
it builds on the basic characterization of $c(x)$, namely,
\beql{a2}
c(x) = \frac{1}{1 - x c(x)} \quad \mbox{or} \quad c(x)^2 = \frac{c(x) - 1}{x}.
\eeq
The first relation in \eqn{a2} characterizes $c(x)$ as the fixed point of the exponential-mixture operator;
see Proposition 4 of \cite{AW10}.
The second relation characterizes the sequence $\{C_n\}$ by having the two-fold convolution equal to $C_{n+1}$.
Based on \eqn{a1} and \eqn{a1a}, we obtain
\beql{a2a}
c(x; \alpha) = \frac{2 \alpha}{2 \alpha - 1 + \sqrt{1 - 4 \alpha x}} = \frac{2 \alpha - 1 - \sqrt{1 - 4 \alpha x}}{2(\alpha - 1 + x)}.
\eeq

The generalized Catalan numbers are given explicitly by
\beql{b3}
C_{n+1} (\alpha) = \sum_{k = 0}^{n} a(n,k) \alpha^k,
\eeq
where the triangle numbers $a(n,k)$ given in \seqnum{A009766} are the famous ballot numbers
\beql{b4}
a(n,k) \equiv \left(1 - \frac{k}{n+1}\right)\binom{n+k}{k};
\eeq
see pp. 130, 152 of \cite{R68}.

\paragraph{Brownian motion and EMIG distributions.}
Brownian motion is one of the most extensively studied stochastic processes; e.g., see Chapter 1 of \cite{H85} or \cite{CM65}.
It has two parameters:  the drift parameter $\mu$ and the diffusion or variance parameter $\sigma^2$.
If $X \equiv \{X(t): t \ge 0\}$ is a $(\mu,\sigma^2)$-Brownian motion, then $X(t)$ has a normal or Gaussian distribution
with mean $\mu t$ and variance $\sigma^2 t$ for each $t$.
We will be interested in the first passage time from one state to another for Brownian motion with drift, which is well understood.
The standard way to study such first passage time problems is to apply martingales.
For the problem at hand with drift, it is standard to apply exponential martingales, in particular, the Wald martingale
\beql{wald}
W(t) \equiv \exp{\{c X(t) - q(c)t\}}, \quad t \ge 0,
\eeq
where $q(c) \equiv \mu c + \sigma^2 c^2/2$; see \S\S 1.5 and 3.2 of \cite{H85}.
We can calculate the Laplace transform of the first passage time distribution by using the optional stopping theorem
with the first passage time serving as the stopping time.

Here we consider Brownian motion (BM) with constant drift $\mu = -1/(2\alpha)$ and constant diffusion coefficient $\sigma^2 = 1/(2 \alpha)$,
depending on the positive real parameter $\alpha$.  By the reasoning above, or in other ways,
we determine the {\em Laplace transform} of the probability density function (pdf) $f(t, y; \alpha)$ of the first passage time from initial state $y &gt; 0$ at time $0$ to final state $0$,
\beql{a3a}
\hat{f} (s, y; \alpha) \equiv \int_{0}^{\infty} e^{-st} f(t,y; \alpha) \, dt  =  \exp{\{- y(\sqrt{1 + 4 \alpha s} - 1)\}},
\eeq
which is defined for complex $s$ with positive real part,
and then the pdf itself,
\beql{a3}
f(t,y; \alpha) = ((\alpha y^2)/(\pi t^3))^{1/2}\exp{\{-(t - 2 \alpha y)^2/(4 \alpha t)\}}, \quad t \ge 0,
\eeq
which is the pdf of an {\em inverse Gaussian} (IG) distribution.
For the classical approach before martingales, see (73) on p. 221 of \cite{CM65}.
  The seminal work was done independently by Schroedinger and Smoluchowski in 1915; see p. 1 of \cite{S93}.

A closely related stochastic process is {\em reflected Brownian motion} (RBM), which is the BM above, with the same parameters, and a reflecting barrier at $0$;
see \S 1.9 and Chapter 5 of \cite{H85}, where it is called regulated Brownian motion (a name which has not caught on).
Clearly, the first-passage-time from $y&gt;0$ to $0$ for RBM has the same pdf $f(t,y; \alpha)$ in \eqn{a3}.  Unlike the BM, the RBM
has a proper limiting distribution as $t \ra \infty$, which is also a stationary distribution for the RBM.  That stationary distribution
has an exponential pdf $p$ with mean $1/2$, i.e.,
\beql{a4}
p(y; \eta) = \eta e^{-\eta y}, \quad y \ge 0, \qforq \eta = 2;
\eeq
see \S5.6 of \cite{H85}.

An important quantity associated with RBM is the first passage time to $0$ starting from the
stationary distribution $p$, which we have called the {\em equilibrium time to emptiness};
see Theorem 1.3 and Corollary 1.3.1 of \cite{AW87}.
Since the equilibrium time to emptiness is a mixture of the distribution of the first-passage time to $0$ with respect
to the initial stationary distribution, where the first-passage-time distribution is inverse Gaussian and the
stationary distribution is exponential,
the equilibrium time to emptiness has a distribution that is an {\em exponential mixture of inverse Gaussian} (EMIG) distributions.



\paragraph{EMIG distributions and the generalized Catalan numbers.}
We will show that a particular family of EMIG pdf's are intimately connected to the generalized Catalan numbers.
We define the family of EMIG pdf's indexed by $\alpha$ as
\beql{a5}
g(t; \alpha) \equiv \int_{0}^{\infty} p(y; \alpha) f(t, y; \alpha) \, dy, \quad t \ge 0.
\eeq
For each $\alpha &gt; 0$, $g(t; \alpha)$ is a bona fide pdf, but it only corresponds to the equilibrium time to emptiness for RBM
in the special case $\alpha = 1$.  However, the EMIG distribution is the first passage time to the origin for both BM and RBM,
when they have negative drift,
with respect to a particular exponential initial distribution on the positive halfline.
We studied EMIG probability distributions in
\cite{AW87,AW95,AW96,AW99a,AW99b}; see \S 2 of \cite{AW95}, \S 8 of \cite{AW96},
Example 8.3 in \cite{AW99a} and \S 3 of \cite{AW99b}.

Now let $\hat{g} (s; \alpha)$ be the Laplace transform of the pdf $g(t; \alpha)$ in \eqn{a5}, i.e.,
\beql{a6}
\hat{g} (s; \alpha) \equiv \int_{0}^{\infty} e^{-s t} g(t; \alpha) \, dt.
\eeq
Let $\hat{g} (-x; \alpha)$ for positive real $x$ be the associated moment generating function of $g(t; \alpha)$.
Our main observation is
\begin{theorem}\label{th1}  The EMIG moment generating function $\hat{g} (-x; \alpha)$
coincides with the generating function $c(x; \alpha)$ in \eqref{a2a}.  Equivalently, the Laplace transform in \eqref{a6}
can be represented as
\beql{a7}
\hat{g} (s; \alpha) = c(-s; \alpha) = \frac{2 \alpha}{2 \alpha - 1 + \sqrt{1 + 4 \alpha s}}.
\eeq
\end{theorem}

\paragraph{Proof.}
By direct integration, we can verify that $\hat{g} (s; \alpha)$ in \eqref{a7} has the integral representation
\beql{b1}
\hat{g} (s; \alpha) = \int_{0}^{\infty} 2 \alpha e^{-2 \alpha y} \hat{f} (s, y; \alpha) \, dy,
\eeq
where $\hat{f} (s, y; \alpha)$ is the Laplace transform in \eqn{a3a};
see (29.3.82) on p. 1026 of \cite{AS72} and (8.4) on p. 95 of \cite{AW96}.~~~\bsq



\section{Further Connections}\label{sec2}



\paragraph{The explicit representation.}

We now apply Theorem \ref{th1} to make connections.
  We first apply Theorem \ref{th1} to give an alternative
  derivation of the explicit expression for the generalized Catalan numbers in \eqn{b3}.  From \eqref{b1}, we directly obtain
\beql{b5}
C_{n+1} (\alpha) = \int_{0}^{\infty} 2 \alpha e^{-2 \alpha y} \frac{m_{n+1} (f)}{(n+1)!} \, dy,
\eeq
where
\beql{moment}
m_n(f) \equiv \int_{0}^{\infty} x^n f(x) \, dx,
\eeq
so that $m_{n+1} (f)$ is the $(n+1)^{\rm st}$ moment of the pdf $f \equiv f(t,y; \alpha)$ in \eqref{a3},
with
\beql{b6}
\frac{m_{n+1} (f)}{(n+1)!} = \sum_{k = 0}^{n} a(n,k) \alpha^k \frac{(2 \alpha y)^{n+1-k}}{(n+1 - k)!};
\eeq
see Proposition 2.14 on p. 46 of \cite{S93}.  After the final expression for $C_{n+1} (\alpha)$ in \eqn{b3} is factored out of \eqn{b5},
the remaining integral reduces to $1$, because it can be identified as the integral of a gamma pdf over its entire domain.

Examples are
\begin{eqnarray}
\{2^n C_n (1/2) \} &amp; = &amp; 1, \, 2, \, 6, \, 20, \, 70, \, 252 \quad (\seqnum{A000984}) \nonumber \\
\{ C_n (2) \} &amp; = &amp; 1, \, 1, \, 3, \, 13, \, 67, \, 381 \quad (\seqnum{A064062}) \nonumber \\
\{2^n C_n (3/2) \} &amp; = &amp; 1, \, 2, \, 10, \, 68, \, 538, \, 4652 \quad (\seqnum{A110520}) \nonumber \\
\{ C_n (3) \} &amp; = &amp; 1, \, 1, \, 4, \, 25, \, 190, \, 1606 \quad (\seqnum{A064063}) \nonumber
\end{eqnarray}
In addition, the OEIS includes $C_n (4)-C_n(10)$ as sequences
\seqnum{A064087}--\seqnum{A064093}.
Note that $\{2^n C_n (1/2)\}$ is the sequence of central binomial numbers (see \eqref{d1} below) and that
$c(x; \alpha) = 1/\sqrt{1 - 2x}$.

It is immediate from \eqref{a1} and the well known continued fraction representation for $c(x)$
that $c(x; \alpha)$ has the continued fraction representation
\beql{cf}
c(x; \alpha) = \frac{1}{1 -} \quad \frac{x}{1 -} \quad  \frac{\alpha x}{1 -} \quad \frac{\alpha x}{1 -} \quad \frac{\alpha x}{1 -}. \cdots
\eeq
As a consequence, we see that $c(x; 0) = (1 - x)^{-1}$.

\paragraph{The generalized Catalan numbers as moments.}

Next we give the mixing-density representation for the EMIG.  We say that a pdf $h (y)$ has a mixing density $w(x)$
if $h$ has the integral representation
\beq
h(t) = \int_{\tau_1}^{\tau_2} y^{-1} e^{-t/y} w(y) \, dy, \quad y \ge 0,
\eeqno
for some fixed $\tau_1$ and $\tau_2$ with $\tau_1 &lt; \tau_2$.
The associated Laplace transforms are related by
\beq
\hat{h}(s) = \int_{\tau_1}^{\tau_2}  w(y) \, \frac{dy}{1 + s y};
\eeqno
see (3.2) and (3.4) in \cite{AW99a}.

For our EMIG pdf's, we have the mixing-density representation
\beql{b7}
 \hat{g} (s; \alpha) = \int_{0}^{4 \alpha} \beta(y; \alpha) \frac{dy}{1 + ys},
 \eeq
 where
 \beql{b8}
 \beta(y; \alpha) \equiv \frac{\sqrt{4 \alpha - y}}{2 \pi \sqrt{y} (1 + (\alpha - 1) y)};
 \eeq
 i.e., $\beta(y; \alpha)$ is the pdf of a generalized beta distribution;
 see (94.22) on p. 375 of \cite{W48} and Theorem 4.1 on p. 29 of \cite{AW99b}.
 For the two cases $\alpha = 1/2$ and $\alpha = 1$, $\beta(y; \alpha)$ is an ordinary beta pdf.

 As an immediate consequence, we obtain the following characterization of the generalized Catalan numbers.
 \begin{theorem}\label{thBetaMoments}
   The generalized Catalan numbers arise as the moments of the generalized beta pdf $\beta (y; \alpha)$ in \eqref{b8};
 i.e.,
 \beql{b9}
 C_n (\alpha) = \int_{0}^{4 \alpha} y^n \beta(y; \alpha) \, dy.
 \eeq
 \end{theorem}
The moments in \eqn{b9} can also be expressed in terms of the Gauss series,
because $C_n (\alpha) = F(1-n, n; -n, a)$; see (15.4.2) on p. 561 of \cite{AS72}.


\section{Implications for Other Integer Sequences}\label{sec3}

\paragraph{A product of EMIG generating functions.}

We now study the (two-fold) convolution of the sequence $\{a^n C_n (b)\}$
with the sequence $\{b^n C_n (a)\}$, i.e., the sequence $\{ C_n (a, b)\}$
defined in \eqn{c1},
using \eqref{a1}.  An interesting property of EMIG distributions is that the
convolution represented by \eqn{c1} can be represented as a linear combination of EMIG's;
see (8.13) on p. 97 of \cite{AW96}.   As a consequence, we obtain

\begin{theorem}\label{th2}
For $a \not= b$,
\beql{c2}
c(bx; a) c(ax; b) = \frac{1}{b-a}\left(b c(bx; a) - a c(ax; b)\right)
\eeq
and
\beql{c3}
C_n (a,b) = \frac{1}{b-a}\left(b^{n+1}C_n (a) - a^{n+1}C_n (b)\right).
\eeq
\end{theorem}

The case $a = b$ is covered by Corollary \ref{corTh3} below.
As an aside, we point out that for $a = b = 1$, the multiple convolutions of $\{C_n\}$ with itself
are represented by the triangle \seqnum{A033184}, which has some very interesting properties; see \cite{L02}.  From
a probabilistic perspective, see p. 568 of \cite{AW87}.

Two examples are
\begin{eqnarray}
\{C_n (2,3) \} &amp; = &amp; 1, \, 5, \, 49, \, 653, \, 10201, \, \cdots \quad (\seqnum{A116873}) \nonumber \\
\{ C_n (2, 4) \} &amp; = &amp; 1, \, 6, \, 76, \, 1336, \, 27696, \,  \cdots \quad (\seqnum{A116874}) \nonumber
\end{eqnarray}
The OEIS includes $\{C_n (2,5)\}$ through $\{C_n (2,8)\}$ as \seqnum{A116875}$(n+1)$ through \seqnum{A116878}$(n+1)$.

To illustrate how Theorem \ref{th2} can be applied, from \eqn{c3} we obtain
\begin{eqnarray}
C_4 (2,3) &amp; = &amp; 3^5 C_4 (2) - 2^5 C_4 (3) \nonumber \\
&amp; = &amp; 3^5 \seqnum{A064062} (4) - 2^5 C_4 (3) \seqnum{A064063} (4) \nonumber \\
&amp; = &amp; 3^5 (67) - 2^5 (190) = 10201. \nonumber
\end{eqnarray}

\paragraph{The sequence $\{V_n (\alpha)\}$.}

We now come to the sequence $\{V_n (\alpha)\}$ in \eqn{int1} and \eqn{d2}.
In \S \ref{sec2} we mentioned that the sequence $\{2^n C_n (1/2)\}$ is the sequence
of central binomial numbers; i.e.,
\beql{d1}
c(2x; 1/2) = \frac{1}{\sqrt{1 - 4x}} = \sum_{n = 0}^{\infty} \binom{2n}{n} x^n.
\eeq
Define the sequence $\{V_n (\alpha)\}$ and its generating function
$v(x; \alpha)$ via \eqn{d2}.
We obtain the following result; we give proofs of this theorem and following ones in \S \ref{secProofs} below.
\begin{theorem}\label{thV}  The numbers $V_n (\alpha)$ have the explicit representation
\beql{d3}
V_n (\alpha) = \frac{1}{2 \alpha - 1}\left( (2 \alpha)^{n+1} C_n (1/2) - C_n (\alpha)\right) = \sum_{k = 0}^{n} \binom{n+k}{k} \alpha^k.
\eeq
\end{theorem}
It seems that the representation in Theorem \ref{thV} is new.
The only two cases of $\{V_n (\alpha)\}$ in the OEIS we are aware of are
\begin{eqnarray}
\{V_n (1) \} &amp; = &amp; 1, \, 3, \, 10, \, 35, \, 126, \, 1716 \, \cdots \quad (
\seqnum{A001700}) \nonumber \\
\{ V_n (2) \} &amp; = &amp; 1, \, 5, \, 31, \, 209, \, 1471, \, 10625 \, \cdots \quad 
(\seqnum{A178792}) \nonumber
\end{eqnarray}
Indeed, we ourselves recently contributed the case $\alpha = 2$.

For the relatively simple case of $\alpha = 1/2$, by \eqn{d2} the generating function is given by $1/(1 - 2x)$, so that
$V_n (1/2) = 2^n$.  Therefore, from Theorem \ref{thV}, we deduce that
\beql{d4}
2^n = \sum_{k=0}^{n} \binom{n+k}{k} \left(\frac{1}{2}\right)^k.
\eeq
Formula \eqn{d4} is given on p. 167 of \cite{GKP94}.%, where it is judged to be ``quite unexpected.''
Also (5.137) on p. 236 of \cite{GKP94} gives the following recurrence relation
\beql{d5}
V_n (\alpha) + (\alpha - 1) V_{n+1} (\alpha) = (2 \alpha - 1) \alpha^{n+1} V_{n} (1).
\eeq

Now consider the convolution of the sequence $\{C_n (\alpha\}$ with itself, denoting its terms by $C_n^{(2)} (\alpha)$; i.e.,
let
\beql{d6}
c^{(2)} (x; \alpha) \equiv \sum_{n=0}^{\infty} C_n^{(2)} x^n \equiv c(x; \alpha)^2.
\eeq

Then we find the following result.

\begin{theorem}\label{th3}  We can represent the numbers $C_n^{(2)} (\alpha)$ via
\beql{d7}
c^{(2)} (x; \alpha) \equiv c(x; \alpha)^2 = 2 \alpha v(x; \alpha) - (2 \alpha - 1) \frac{d}{dx} c(x; \alpha)
\eeq
and
\beql{d8}
C^{(2)}_n (\alpha) = 2\alpha V_n (\alpha) - (2\alpha - 1) (n+1) C_{n+1} (\alpha).
\eeq
\end{theorem}

Two examples are
\begin{eqnarray}
\{C_n^{(2)} (2) \} &amp; = &amp; 1, \, 2, \, 7, \, 32, \, 169, \,  \cdots \quad (\seqnum{A115197}) \nonumber \\
\{ C_n^{(2)} (3) \} &amp; = &amp; 1, \, 2, \, 9, \, 58, \, 446, \, \cdots \quad (\seqnum{A116867}) \nonumber
\end{eqnarray}

We can apply Theorem \ref{th3} to treat the missing case in Theorem \ref{th2}; i.e., we
can determine the numbers $C_n (a, a)$.

\begin{corollary}\label{corTh3}  We can represent the numbers $C_n (a, a)$ via
\beql{d9}
C_n (a, a) = a^n C_n^{(2)} (a).
\eeq
\end{corollary}

By combining \eqn{d8} and \eqn{d9}, we can obtain explicit representations for the sequence $\{C_n (a,a)\}$.
The OEIS includes $\{C_n (2,2)\}$ through $\{C_n (9,9)\}$ as \seqnum{A064340} (n+1) through \seqnum{A064347} (n+1).

The following integral representation for $V_n (\alpha)$ is obtained from Theorem \ref{thV} using \eqn{b7}-\eqn{b9}
\begin{theorem}\label{thIntRepV}
 We have the following integral representation for the numbers $V_n (\alpha)$
 \beql{d10}
V_n (\alpha)  =  \int_{0}^{4 \alpha} y^n w(y; \alpha) dy,
\eeq
where the mixing density is given by
\begin{eqnarray}\label{d10a}
w(y; \alpha) &amp; = &amp; \frac{1}{2 \alpha -1} \left( \frac{2 \alpha}{\pi \sqrt{y (4 \alpha - y)}} - \beta(y; \alpha)\right) \nonumber \\
&amp; = &amp; \frac{1}{2 \pi}\sqrt{y/(4 \alpha - y)}\left(\frac{2 \alpha - 1}{1 + (\alpha - 1) y}\right).
\end{eqnarray}
\end{theorem}
Paralleling the observation after Theorem \ref{thBetaMoments}, we note that
the numbers $V_n (\alpha)$ also have a Gauss series representation, namely,  $V_n (\alpha) = F(-n, n+1; -n; \alpha)$.
We discuss it further below.

Next we give a continued fraction representation for $v(x; \alpha)$.

\begin{theorem}\label{thVcf}  The generating function $v(x; \alpha)$ has the continued fraction representation
\beql{d11}
v(x; \alpha) = \frac{1}{1 - (2\alpha + 1)x -} \, \, \frac{\alpha (2\alpha - 1)x^2}{1 - 2\alpha x -} \, \,
\frac{(\alpha x)^2 }{1 - 2\alpha x -} \, \, \frac{(\alpha x)^2 }{1 - 2\alpha x -} \, \, \cdots
\eeq
\end{theorem}
Note that $v(x,0) = 1/(1 - x)$.  From Theorem \ref{thVcf}, we can ``pick off'' the Hankel transform of the integer sequence $\{V_n (\alpha)\}$.
As in \cite{AW10,B09},
the Hankel transform of an integer sequence provides a useful partial characterization;
it is a many-to-one function mapping an integer sequence into another integer sequence.
Starting from
a sequence $\{\omega_n: n \ge 0 \} \equiv \omega_0, \omega_1, \omega_2, \omega_2, \dots$
with $\omega_0 \equiv 1$, let the
{\em Hankel matrix} $M^{(n)}$ be the $(n+1) \times (n+1)$ symmetric matrix with elements $M^{(n)}_{i,j} \equiv \omega_{i+j-2}$,
$0 \le i \le n$, $0 \le j \le n$.
(The first row contains the first $n+1$ elements and $M_{n+1,n+1} \equiv \omega_{2n}$.
Let $H_{2n} \equiv det(M^{(n)})$, the {\em even  Hankel determinant}.
Let the {\em Hankel transform} of the sequence $\{\omega_n: n \ge 0 \}$ above be the sequence
$\{H_{2n}: n \ge 0\}$; it starts with $H_0 = 1$.
With \eqn{d11}, we can apply (12.2) and (12.3) of \cite{B09} to obtain
\beql{Hankel}
HT(\{V_n (\alpha\}) = (2\alpha - 1)^n \alpha^{n^2}.
\eeq

\paragraph{The Gauss Contiguous Relation.}
After Theorems \ref{thBetaMoments} and \ref{thIntRepV}, we observed that both $C_n (\alpha)$ and $V_n (\alpha)$
have Gauss series representations, namely,
\beql{g1}
C_n (\alpha) = F(1-n, n; -n; \alpha) \qandq V_n (\alpha) = F(-n, n+1; -n; \alpha).
\eeq
These are nicely linked via the Gauss contiguous relation (15.2.14) on p. 558 of \cite{AS72}, yielding
\beql{g2}
\frac{C_n (\alpha)}{2} + \frac{V_n (\alpha)}{2} = F(-n,n; -n; \alpha) \equiv \sum_{k=0}^{\infty} R(n,k) \alpha^k,
\eeq
where
\beql{g3}
R(n,k) \equiv \binom{n+k-1}{k} = \binom{-n}{k} (-1)^k,
\eeq
as in \seqnum{A158498}.  The case $\alpha = 2$ is \seqnum{A119259}.

\section{Connections with Our Previous Papers}\label{secPapers}

Our interest was drawn to the generating functions in \eqn{a1}-\eqn{d2} largely because they are natural generalizations
of probabilistic quantities that we studied previously via Laplace transforms.  These generating functions in \eqn{a1}-\eqn{d2} are directly moment generating functions (mgf's)
of probability density functions on the nonnegative halfline.  If we replace $x$ by $-s$, where $s$ is a complex number with positive real part, then we obtain the corresponding
Laplace transform.  The relations we have established are {\em simultaneously} relations among integer sequences and relations among probability distributions.  We think that
we have uncovered relations of interest in {\em both} domains.  We are also intrigued by the possibility of establishing more connections between the two domains.

First, as noted before Theorem 9 of \cite{AW10}, $c(-s) = \hat{h}_1 (s)$, where $\hat{h}_1 (s)$ is the Laplace transform of the pdf $h_1 (t)$ of the first moment cdf of RBM,
$H_1 (t) \equiv E[R(t)|R(0) = 0]/E[R(\infty)]$, $t \ge 0$, which we first studied in \cite{AW87}.  In \cite{AW87} we consider RBM with drift coefficient $-1$ and diffusion coefficient $1$,
whereas here we consider RBM with drift coefficient $-1/(2\alpha)$ and diffusion coefficient $1/(2\alpha)$, so that there is a scale difference, even when $\alpha = 1$;
e.g., in \cite{AW87}, $\hat{h}_1 (s) = c(-s/2)$ for $c(x)$ in \eqn{a1}.

We can expand upon the second relation in \eqn{a2}.  Corollaries 1.3.2 and 1..5.1 in \cite{AW87}
show that the Laplace transform of the pdf $h_2$ of the RBM second moment cdf $H_2 (t) \equiv E[R(t)^2|R(0)=0]/E[R(\infty)^2]$
can be represented as
\beql{second}
\hat{h}_2 (s) \equiv \int_{0}^{\infty} e^{-ys} h_2 (y) \, dy = \hat{h}_{1,e} (s) \equiv \frac{1 - \hat{h}_1 (s)}{s} = \hat{h}_1 (s)^2,
\eeq
where, for any cdf $H (t) \equiv P(X \le t)$, the associated stationary-excess (or equilibrium residual lifetime) cdf and pdf are defined by
\beql{excess}
H_e (t) \equiv \frac{1}{E[X]} \int_{0}^{t} P(X &gt; y) \, dy \qandq h_e (t) \equiv \frac{1-H(t)}{E[X]}, \quad t \ge 0,
\eeq
having associated Laplace transform
\beql{transforms}
\hat{h}_e (s) \equiv \int_{0}^{\infty} e^{-sy} h_e (y) \, dy =  \frac{1 - \hat{h}(s)}{E[X] s}.
\eeq
Thus, equation \eqn{second} states that the RBM second moment pdf $h_2$ is simultaneously the stationary excess of the RBM first-moment pdf $h_1$
and the two-fold convolution of the cdf $h_1$ with itself.  The pdf $h_1$ is the only pdf on the nonnegative real line for which
the associated stationary-excess pdf coincides with the two-fold convolution..  As discussed in Theorem 9 of \cite{AW10}, the pdf $h_1$ is intimately connected to the
Catalan numbers in \eqn{a1}.

Next, the definition we use for the generalized Catalan numbers in \eqn{a1a} is tantamount to applying the exponential mixture operator
to construct a new probability distribution via its Laplace transform.  As discussed in \S 7 of \cite{AW96}, given a Laplace transform $\hat{f}$ of a pdf $f$,
the exponential-mixture operator yields $\sE\sM (\hat{f})(s) \equiv (1 + s\hat{f}(s))^{-1}$.  We get the two scale parameters by first replacing $\hat{f} (s)$
by $\hat{f} (a s)$ for some $a &gt; 0$ and then replacing $\sE\sM (\hat{f})(s)$ by $\sE\sM (\hat{f})(b s)$ for some $b &gt; 0$.  With the introduction of these two scale parameters,
we see that $c(-s; \alpha)$ can be viewed as an application of the exponential-mixture operator, so that it too is the Laplace transform of a bona fide pdf.
Next, the product operation in \eqn{c1} and \eqn{d2} is known to correspond to convolution,
so that all generating functions correspond directly to mgf's of probability density functions.

Finally, the new relations in \eqn{c1} and \eqn{d2} generalize previous ones in our earlier papers.
First, when $a = b = 1$, equation \eqn{c1} is equivalent to the equivalence of the two representations for the RBM second moment pdf $h_2$ in \eqn{second} above.
When $\alpha = 1$, the relation in \eqn{d2}, appropriately adjusted for scale,
reduces to a special relation among distributions that are Beta mixtures of exponential distributions, as defined in \cite{AW99c}.
In the language of \cite{AW99c},
\beql{last}
\hat{v}_e (1/2,1/2; s) = \hat{v} (3/2,1/2; s) = \hat{v} (1/2,1/2; s) \cdot \hat{v} (1/2,3/2; s)
\eeq
where
\beql{Vdef}
v(p,q;t) \equiv \int_{0}^{1} y^{-1} e^{-ty} b(p, q; y) \, dy \qandq b(p, q; y) \equiv \frac{\Gamma(p + q)}{\Gamma(p)\Gamma(q)} y^{p-1} (1-y)^{q-1}
\eeq
and the subscript $e$ again corresponds to the stationary-excess operator in \eqn{excess} above.
The first equality in \eqn{last} follows from (1.14) in Theorem 1.3 in \cite{AW99c}.
By Table 3 on p. 536 of \cite{AW99c}, the second relation in \eqn{last} then reduces to
\beql{last2}
\hat{\gamma}_e (s) = \hat{\gamma}(s) \cdot \hat{h}_1 (s),
\eeq
where
\beql{gamma}
\gamma (t) \equiv \gamma (1/2; t) \equiv \frac{e^{-t}}{\sqrt{\pi t}} \qandq \hat{\gamma} (s) \equiv \hat{\gamma} (1/2; s) = \frac{1}{\sqrt{1 + s}}.
\eeq
The second relation in \eqn{last} and the equivalent representation in \eqn{last2}
can be verified by direct calculation from the explicit expressions above and in Table 3 of \cite{AW99c}.

And all of this has a corresponding story in the world of integer sequences.  In particular,
the associated integer sequences are:  for $\gamma (1/2;t)$ in \eqn{last2} and \eqn{gamma}: $C(2n,n) \equiv (1, 2, 6, 20, 70, \ldots$ (\seqnum{A000984}),
for $h_1$:  $C_n \equiv (1, 1, 2, 5, 14, \ldots)$ in \eqn{int1a} (\seqnum{A000108}); and for $\gamma_e$ in \eqn{last2}: $C(2n+1,n) \equiv (1, 3, 10, 35, 126, \ldots)$ (\seqnum{A001700}).
Our equations \eqn{d2} and \eqn{d3} generalize
\begin{eqnarray}
\seqnum{A001700} &amp; = &amp; \mbox{convolution}(\seqnum{A000984}, \seqnum{A000108}) \nonumber \\
\seqnum{A001700} &amp; = &amp; 2\cdot \seqnum{A000984} - \seqnum{A000108}, \nonumber
\end{eqnarray}
occurring when $\alpha = 1$.  The second formula above is not yet in the formula section for \seqnum{A001700}.

In summary, our goal here, as in our previous papers \cite{AW94,AW96,AW99c,AW10}
is to generalize the relations originally observed in \cite{AW87} and, at the same time, make connections to integer sequences.
We view this paper as uncovering a little bit more about a much bigger story, rather than just generating a few sequences not yet in the OEIS \cite{S10}..
A major part of that larger story is the development of an operational calculus for probability distributions, which may be implemented
via their transforms and moment sequences as well as via the pdf's and cdf's..
This short paper is neither the end of that story nor the beginning.


\section{Proofs}\label{secProofs}

\paragraph{Proof of Theorem \ref{thV}.}
Multiply the recurrence \eqn{d5} by $x^n$ and sum, producing
\beq
\sum_{n=0}^{\infty} V_n (\alpha) x^n + (\alpha -1) \sum_{n=0}^{\infty} V_{n+1} (\alpha) x^n
= (2\alpha -1) \alpha \sum_{n=0}^{\infty} V_n (1) (\alpha x)^n.
\eeqno
Apply that to get a relation for the generating functions, namely,
\beq
v(x; \alpha) + \frac{(\alpha - 1) (v(x; \alpha) - 1)}{x} = \frac{(2 \alpha -1 )(1 - \sqrt{1 - 4 \alpha x})}{2x \sqrt{1 - 4 \alpha x}},
\eeqno
which yields
\beq
v(x; \alpha) = \frac{2 \alpha -1 - \sqrt{1 - 4 \alpha x}}{2(x+ \alpha -1)\sqrt{1 - 4\alpha x}},
\eeqno
which is equivalent to \eqn{d3}.

\paragraph{Proof of Theorem \ref{th3}.}
Observe that
\begin{eqnarray}
2 \alpha v(x; \alpha) - (2 \alpha -1) \frac{d}{dx} c(x; \alpha)
&amp; = &amp; \frac{(2 \alpha)^2}{\sqrt{1 - 4 \alpha x} ( 2 \alpha -1 + \sqrt{1 - 4 \alpha x})}  \nonumber \\
&amp;&amp; \quad \quad - \frac{(2 \alpha -1 )(2 \alpha)^2}{\sqrt{1 - 4 \alpha x} (2 \alpha - 1 + \sqrt{1 - 4 \alpha x})^2} \nonumber \\
&amp; = &amp; c(x; \alpha)^2 \left(\frac{2 \alpha -1 + \sqrt{1 - 4 \alpha x}}{\sqrt{1 - 4 \alpha x}} - \frac{2 \alpha -1}{\sqrt{1 - 4 \alpha x}}\right) \nonumber \\
&amp; = &amp; c(x; \alpha)^2.  \nonumber
\end{eqnarray}


\paragraph{Proof of Theorem \ref{thIntRepV}.}
As a consequence of Theorem \ref{thV},
\beq
w(y; \alpha) = \frac{2 \alpha \phi(y; \alpha) - \beta(y; \alpha)}{2 \alpha - 1},
\eeqno
where $\phi (y; \alpha)$ is the mixing density of $c(2 \alpha x; 1/2)$ and  $\beta (y; \alpha)$ is the mixing density
of $c(x;\alpha)$ given in \eqn{b8}.  From \eqn{d1}, $c(2 \alpha x; 1/2) = 1/\sqrt{1 - 4 \alpha x}$.  Therefore,
$\phi(y; \alpha) = 1/\pi \sqrt{y(4 \alpha - y)}$.  Hence, we have
\begin{eqnarray}
w(y; \alpha) &amp; = &amp; \frac{1}{2 \alpha - 1}\left(\frac{2 \alpha}{\pi \sqrt{y(4 \alpha - y)}} - \frac{\sqrt{4 \alpha - y}}{2 \pi \sqrt{y} (1 + (\alpha - 1) y)}\right) \nonumber \\
&amp; = &amp; \left(\frac{1}{2 \pi (2 \alpha -1)}\right) \left(\frac{\sqrt{y}}{\sqrt{4 \alpha - y}}\right) \left(\frac{4 \alpha}{y} - \frac{4 \alpha - y}{y(1 + (\alpha - 1)y)}\right), \nonumber
\end{eqnarray}
which gives the desired result \eqn{d10a}.

\paragraph{Proof of Theorem \ref{thVcf}.}
Let $Q$ be the continued fraction
\beq
Q \equiv \frac{(\alpha x)^2 }{1 - 2\alpha x -} \, \, \frac{(\alpha x)^2 }{1 - 2\alpha x -} \, \, \frac{(\alpha x)^2 }{1 - 2\alpha x -} \, \, \cdots
\eeqno
Then Theorem \ref{thVcf} is equivalent to
\beql{m1}
v(x; \alpha) = \frac{1}{1 - (2 \alpha + 1) x - (2\alpha - 1) Q/\alpha}.
\eeq
Formula \eqn{m1} is proved by showing that
\beql{m2}
(i) \, \, Q = \alpha x(c(\alpha x) - 1) \qandq (ii) \, \,
v(x; \alpha) = \frac{1}{(1 - 2x - (2\alpha -1)x c(\alpha x))}.
\eeq

Proof of (i) in \eqn{m2}:  From
\beq
c(\alpha x) = \frac{1}{1-} \, \, \frac{\alpha x}{1 -} \, \, \frac{\alpha x}{1 -} \, \,  \cdots
\eeqno
we have the even part as
\begin{eqnarray}
c(\alpha x) &amp; = &amp; \frac{1}{1-\alpha x -} \, \, \frac{(\alpha x)^2}{1 -2 \alpha x -} \, \, \frac{(\alpha x)^2}{1 -2 \alpha x -} \, \,  \cdots \nonumber \\
&amp; = &amp; \frac{1}{1 - \alpha x - Q} .\nonumber
\end{eqnarray}
Hence, using \eqn{a2} via $1/c(x) = 1 - x c(x)$, we obtain
\beq
Q = 1- \alpha x - \frac{1}{c(\alpha x)} = \alpha x (c(\alpha x) - 1).
\eeqno

Proof of (ii) in \eqn{m2}:  From \eqn{d2}, $v(x; \alpha) = c(2x; 1/2)c(x; \alpha)$
and from \eqn{a1}, \eqn{a1a} and \eqn{a2a}, we find that
\beq
v(x; \alpha) = \left(\frac{1}{1 - 2\alpha x c(\alpha x)}\right)\left(\frac{1}{1 - x c(\alpha x)}\right).
\eeqno
Then after some algebra, exploiting \eqn{a2} via $x c(x)^2 = c(x) - 1$, we get the desired result.

\section{Acknowledgment}

The second author was supported by NSF Grant CMMI 0948190.


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J. Abate and W. Whitt, Limits and approximations for the busy-period distribution in single-server queues,
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J. Abate and W. Whitt, An operational calculus for probability distributions via Laplace transforms,
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J. Abate and W. Whitt, Computing Laplace transforms for numerical inversion via continued fractions,
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J. Abate and W. Whitt, Explicit $M/G/1$ waiting-time distributions for a class of long-tail service-time distributions,
{\em Oper. Res. Letters} {\bf 25} (1999), 25--31.

\bibitem{AW99c}
J. Abate and W. Whitt,
Modeling service-time distributions with non-exponential tails: Beta mixtures of exponentials.
{\em Stochastic Models} {\bf 15} (1999), 517--546.

\bibitem{AW10}
J. Abate and W. Whitt, Integer sequences from queueing theory,
{\em J. Integer Sequences} {\bf 13} (2010), Article 10.5.5.

\bibitem{AS72}
M. Abramowitz and I. Stegun,
{\em Handbook of Mathematical Functions},
Dover, New York.

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(2006), Article 06.2.4.

\bibitem{B09}
P. Barry, {\em A Study of Integer Sequences, Riordan Arrays, Pascal-like Arrays and Hankel Transforms},
Ph.\ D.\ dissertation,
Department of Mathematics, University College Cork, Ireland, 2009.

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D. R. Cox and H. D. Miller,
{\em The Theory of Stochastic Processes}, Methuen, London, 1965.


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R. L. Graham, D. E. Knuth, and O. Patashnik,
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V. Seshadri,
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\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}:
Primary 11B25; Secondary 60J65, 60E05.

\noindent \emph{Keywords: }  Catalan numbers, generalized Catalan numbers,
inverse Gaussian probability distributions, exponential mixtures,
exponential mixtures of inverse Gaussian distributions, Brownian motion,
generalized beta distribution,  equilibrium time to
emptiness, moments.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences 
\seqnum{A000108},
\seqnum{A000984},
\seqnum{A001700},
\seqnum{A006633},
\seqnum{A009766},
\seqnum{A033184},
\seqnum{A064062},
\seqnum{A064063},
\seqnum{A064087},
\seqnum{A064088},
\seqnum{A064089},
\seqnum{A064090},
\seqnum{A064091},
\seqnum{A064092},
\seqnum{A064093},
\seqnum{A064340},
\seqnum{A064341},
\seqnum{A064342},
\seqnum{A064343},
\seqnum{A064344},
\seqnum{A064345},
\seqnum{A064346},
\seqnum{A064347},
\seqnum{A068765},
\seqnum{A110520},
\seqnum{A115197},
\seqnum{A116867},
\seqnum{A116873},
\seqnum{A116874},
\seqnum{A116875},
\seqnum{A116876},
\seqnum{A116877},
\seqnum{A116878},
\seqnum{A119259},
\seqnum{A130564},
\seqnum{A158498}, and
\seqnum{A178792}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received August 6 2010;
revised version received   December 5 2010; February 8 2011.
Published in {\it Journal of Integer Sequences}, February 20 2011.

\bigskip
\hrule
\bigskip

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