On symmetry of discrete polynomial hypergroups

Abstract:

Let $H$ be a discrete polynomial hypergoup on $\mathbb {N}$ with Plancherel measure $\mu .$ If the hypergroup $H$ is symmetric, the set of characters $\widehat {H}$ can be identified with a compact subset of the real line which contains the support of $\mu .$ We show that the lower and upper bounds of $\widehat {H}$ and $\operatorname {supp} \mu$ coincide. In particular, the trivial character belongs to the support of the Plancherel measure.