On symmetry of discrete polynomial hypergroups

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 $\supp \mu$ coincide. In particular, the trivial character belongs to the support of the Plancherel measure.