Brownian Motion and the Generalized Catalan Numbers

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.