Brownian Motion and the Generalized Catalan Numbers
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Ridgewood, NJ 07450-2908
Department of Industrial Engineering and Operations Research
New York, NY 10027-6699
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.
Full version: pdf,
(Concerned with sequences
Received August 6 2010;
revised version received December 5 2010; February 8 2011.
Published in Journal of Integer Sequences, February 20 2011.
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