  The block counting process and the fixation line of the Bolthausen-Sznitman
coalescent are analyzed. Spectral decompositions for their generators and
transition probabilities are provided leading to explicit expressions for
functionals such as hitting probabilities and absorption times. It is
furthermore shown that the block counting process and the fixation line of the
Bolthausen-Sznitman $n$-coalescent, properly scaled, converge in the Skorohod
topology to the Mittag-Leffler process and Neveu's continuous-state branching
process respectively as the sample size $n$ tends to infinity. Strong relations
to Siegmund duality and to Mehler semigroups and self-decomposability are
pointed out.
