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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. [ABSTRACT FROM AUTHOR]