  This paper derives explicit results for the infinite Gibbs partitions
generated by the jumps of an $\alpha-$stable subordinator, derived in Pitman
\cite{Pit02, Pit06}. We first show that for general $\alpha$ the conditional
EPPF can be represented as ratios of Fox-$H$ functions, and in the case of
rational $\alpha,$ Meijer-G functions. Furthermore the results show that the
resulting unconditional EPPF's, can be expressed in terms of H and G transforms
indexed by a function h. Hence when h is itself a H or G function the EPPF is
also an H or G function. An implication, in the case of rational $\alpha,$ is
that one can compute explicitly thousands of EPPF's derived from possibly
exotic special functions. This would also apply to all $\alpha$ except that
computations for general Fox functions are not yet available. However, moving
away from special functions, we demonstrate how results from probability theory
may be used to obtain calculations. We show that a forward recursion can be
applied that only requires calculation of the simplest components. Additionally
we identify general classes of EPPF's where explicit calculations can be
carried out using distribution theory.
