  We determine the representation of the group of automorphisms for cyclotomic
function fields in characteristic $p > 0$ induced by the natural action on the
space of holomorphic differentials via construction of an explicit basis of
differentials. This includes those cases which present wild ramification and
automorphism groups with non-cyclic $p$-part, which have remained elusive. We
also obtain information on the gap sequences of ramified primes. Finally, we
extend these results to rank one Drinfel'd modules.
