  An important line of research is the investigation of the laws of random
variables known as Dirichlet means as discussed in Cifarelli and
Regazzini(1990). However there is not much information on inter-relationships
between different Dirichlet means. Here we introduce two distributional
operations, which consist of multiplying a mean functional by an independent
beta random variable and an operation involving an exponential change of
measure. These operations identify relationships between different means and
their densities. This allows one to use the often considerable analytic work to
obtain results for one Dirichlet mean to obtain results for an entire family of
otherwise seemingly unrelated Dirichlet means. Additionally, it allows one to
obtain explicit densities for the related class of random variables that have
generalized gamma convolution distributions, and the finite-dimensional
distribution of their associated L\'evy processes. This has implications in,
for instance, the explicit description of Bayesian nonparametric prior and
posterior models, and more generally in a variety of applications in
probability and statistics involving Levy processes.
