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 expansions as the sample size $n$ tends to infinity for the moments of the total external branch length of the Bolthausen--Sznitman coalescent. The proof is based on an elementary difference method. An alternative differential equation method is developed which can be used 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 $n\ge 4$.