Gibbs Partitions, Riemann-Liouville Fractional Operators, Mittag-Leffler Functions, and Fragmentations Derived From Stable Subordinators

Pitman (2003) (and subsequently Gnedin and Pitman (2006)) showed that a large class of random partitions of the integers derived from a stable subordinator of index $\alpha\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. In particular, the Brownian case $\alpha=1/2,$ may be expressed in terms of Hermite functions. We, HJL (2007), for $\alpha\in(0,1),$ showed that the relevant quantities are Fox $H$ and Meijer $G$ functions, thus in principle allowing for the calculation of a myriad of partition distributions. The most notable member of this class are the $(\alpha,\theta)$ partitions, derived from mass partitions having a $\mathrm{PD}(\alpha,\theta)$ distribution, which are induced by mixing over variables with generalized Mittag-Leffler distributions, denoted by $\mathrm{ML}(\alpha,\theta).$ We provide further interpretations of the broader class. We start with representations in terms of Riemann-Liouville fractional integrals indexed by a stable density. This leads to connections to fractional calculus, wherein the interplay between special functions and probability theory, in particular as it relates to size biased sampling, is illustrated. A centerpiece of our work are results related to Mittag-Leffler functions which plays a key role in fractional calculus. Leading to connections to a mixed Poisson waiting time framework. We provide novel characterizations of general laws related to two nested families of $\mathrm{PD}(\alpha,\theta)$ mass partitions appearing in the literature, by Dong, Goldschmidt and Martin (2005) and Pitman (1999), that exhibit dual coagulation/fragmentation relations, constitute Markov chains, and are otherwise connected to the construction of various random trees and graphs. Simplifications in the Brownian case are also highlighted, indicating relations in the current literature.