It is well known that all commutative hypergroups are amenable in the sense that an invariant mean exists. However, in contrast to the group case, there exist commutative hypergroups for which other versions of amenability fail. For instance, it is well-understood when Reiterâ€™s condition (P2) fails. In the paper under review, the amenability of the Banach-

$*$-algebra

${l}^{1}$ is studied for polynomial hypergroups. It turns out that most known polynomial hypergroups are not amenable in this sense, and that the only known positive example appears for the hypergroup associated with Chebychev polynomials of the first kind.