  A recursion for the joint moments of the external branch lengths for
coalescents with multiple collisions ($\Lambda$-coalescents) is provided. This
recursion is used to derive asymptotic results as the sample size $n$ tends to
infinity for the joint moments of the external branch lengths and for the
moments of the total external branch length of the Bolthausen-Sznitman
coalescent. These asymptotic results are based on a differential equation
approach, which is as well useful to obtain exact solutions for the joint
moments of the external branch lengths for the Bolthausen-Sznitman coalescent.
The results for example show that the lengths of two randomly chosen external
branches are positively correlated for the Bolthausen--Sznitman coalescent,
whereas they are negatively correlated for the Kingman coalescent provided that
$n\ge 4$.
