Krafft and Schaefer [14] considered a two-parameter Ehrenfest urn model and found the n-step transition probabilities using representations by Krawtchouk polynomials. For a special case of the model Palacios [17] calculated some of the expected first-passage times. This note investigates a generalization of the two-parameter Ehrenfest urn model where the transition probabilities pi,i+1 and pi,i+1 are allowed to be quadratic functions of the current state i. The approach used in this paper is based on the integral representations of Karlin and McGregor [9] and can also be used for Markov chains with an infinite state space.
