Coagulation fragmentation laws induced by general coagulations of two-parameter Poisson-Dirichlet processes

Pitman~(1999) describes a duality relationship between fragmentation and coagulation operators. An explicit relationship is described for the two-parameter Poisson-Dirichlet laws, with parameters {\footnotesize (α,θ)} and (β,θ/α), wherein PD(α,θ) is coagulated by PD(β,θ/α) for 0<α<1, 0≤β<1 and −β<θ/α. This remarkable explicit agreement was obtained by combinatorial methods via exchangeable partition probability functions~(EPPF). This work discusses an alternative analysis which can feasibly extend the characterizations above to more general models of PD(α,θ) coagulated with some law Q. The analysis exploits distributional relationships between compositions of species sampling random probability measures and coagulation operators and recent work on Cauchy-Stieltjes transforms of random probability measures by Vershik, Yor and Tsilevich (2004) and James (2002). We use this to obtain explicit descriptions in the case where {\footnotesize Q} corresponds to a large class of power tempered Poisson Kingman models analyzed in James~(2002). That is, explicit results are obtained for models outside of the PD(β,θ/α) family.