Stirling's approximations for exchangeable Gibbs weights

We obtain some approximation results for the weights appearing in the exchangeable partition probability function identifying Gibbs partition models of parameter $\alpha \in (0,1)$, as introduced in Gnedin and Pitman (2006). We rely on approximation results for central and non-central generalized Stirling numbers and on known results for conditional and unconditional $\alpha$ diversity. We provide an application to an approximate Bayesian nonparametric estimation of discovery probability in species sampling problems under normalized inverse Gaussian priors.