(Received April 11, 1996; revised January 10, 1997)
Abstract. An urn has balls of colors C1 and C2. It is replenished (R) by balls of both colors and then depleted by (D) the same number; this constitutes a cycle. When R = D, the system is closed and equilibrium will be reached after many cycles. The ultimate distribution is found only when the replenishment is the same for each color. Asymptotic normal and asymptotic binomial distributions arise when the parameters reach extreme values. For the multicolor urn an expression is given for the correlation between the number of balls of any two colors.
Key words and phrases: Bernard's urn, beta integral transforms, finite difference calculus, generating functions, hypergeometric distributions, hypergeometric functions, moments, replenishment-depletion urn.