A Spectral decomposition for the Bolthausen-Sznitman coalescent and the Kingman coalescent

We consider both the Bolthausen-Sznitman and the Kingman coalescent restricted to the partitions of $\{1,\dots ,n\}\mathrm{.}$ Spectral decompositions of the corresponding generators are derived. As an application we obtain a formula for the Green’s functions and a short derivation of the well-known formula for the transition probabilities of the Bolthausen-Sznitman coalescent.