Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.6

Brownian Motion and the Generalized Catalan Numbers

Joseph Abate
900 Hammond Road
Ridgewood, NJ 07450-2908

Ward Whitt
Department of Industrial Engineering and Operations Research
Columbia University
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.

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

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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