Multiple Hypergeometric Functions: Probabilistic Interpretations and Statistical Uses
综述多重超几何函数的数学性质，将其表示为混合多项分布的概率生成函数，并开发推广的狄利克雷分布族用于多项抽样和列联表的贝叶斯推断，还给出二次型矩的新积分公式。
Abstract This article reviews and interprets recent mathematics of special functions, with emphasis on integral representations of multiple hypergeometric functions. B.C. Carlson's centrally important parameterized functions R and ℛ, initially defined as Dirichlet averages, are expressed as probability-generating functions of mixed multinomial distributions. Various nested families generalizing the Dirichlet distributions are developed for Bayesian inference in multinomial sampling and contingency tables. In the case of many-way tables, this motivates a new generalization of the function ℛ. These distributions are also useful for the modeling of populations of personal probabilities evolving under the process of inference from statistical data. A remarkable new integral identity is adapted from Carlson to represent the moments of quadratic forms under multivariate normal and, more generally, elliptically contoured distributions. This permits the computation of such moments by simple quadrature.