Matching the finitized Poisson distribution to the matching distributions

In this paper, we show that the families of finitized Poisson and generalized matching distributions coincide. This somewhat surprising fact provides for simplified characterizations of generalized matching distributions that were not previously known. A result on convolutions and a reproductive-type property for finitized Poisson distributions, and hence generalized matching distributions, is also presented and illustrated with an example applied to a generalized matching problem.