Summary: [For the entire collection see Zbl 0708.00015.]
A variety of parameterized univariate and vector-variate distributions are generalized by multiplying the usual density by a generalized hypergeometric function or a multiple hypergeometric function with an appropriately transformed argument variable. A benefit of our choice of the new special-function factor is that the normalizing constant has a closed functional form. Also, the moments and characteristic functions of the generalized distributions are averages of the moments and characteristic functions of the usual distributions. The new random quantities, in the cases of the multiple gamma, Dirichlet, and inverted Dirichlet, obey functional relations similar to those of the usual random quantities. The new generalized chi-squared distribution includes, as a special case, the usual noncentral chi-squared distribution. The method preserves the property of being a Bayesian conjugate prior density.