The classical and extended occupancy distributions are useful for examining the number of occupied bins in problems involving random allocation of balls to bins. We examine the extended occupancy problem by framing it as a Markov chain and deriving the spectral decomposition of the transition probability matrix. We look at three distributions of interest that arise from the problem, all involving the noncentral Stirling numbers of the second kind. These distributions give a useful generalisation to the binomial and negative-binomial distributions. We examine how these distributions relate to one another, and we derive recursive properties and mixture properties that characterise the distributions.

翻译：经典和扩展占有分布对于研究随机分配球到箱的问题中占有箱的数量具有重要价值。我们通过将扩展占有问题构建为马尔可夫链，并推导转移概率矩阵的谱分解来研究该问题。我们关注该问题衍生的三种重要分布，这些分布均涉及第二类非中心斯特林数。这些分布为二项分布和负二项分布提供了有用的推广。我们探讨了这三种分布之间的相互关系，并推导了表征这些分布的递归性质与混合性质。