A Markovian Perspective on the Classical Occupancy Problem with a Generalization to Pure Birth Processes

We study the classical occupancy problem from the viewpoint of its embedding Markov chain. We derive new expressions for the probability mass function and (complementary) distribution function in generalized form. Furthermore, we derive a completely novel sparse bidiagonal system of recursion relations for the (complementary) distribution function and provide its efficient matrix implementation. Importantly, we generalize these results to the entire class of discrete-time pure birth processes.