  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.
