  Different notions of amenability on hypergroups and their relations are
studied. Developing Leptin's theorem for discrete hypergroups, we characterize
the existence of a bounded approximate identity for hypergroup Fourier
algebras. We study the Leptin condition for discrete hypergroups derived from
the representation theory of some classes of compact groups. Studying
amenability of the hypergroup algebras for discrete commutative hypergroups, we
obtain some results on amenability properties of some central Banach algebras
on compact and discrete groups.
