  Let $H$ be an ultraspherical hypergroup associated with a locally compact
group $G$ and a spherical projector $\pi$ and let $A(H)$ and $B(H)$ denote the
Fourier and Fourier-Stieltjes algebras, respectively, associated with $H.$ In
this note, we study power boundedness and Ces\`aro boundedness in $B(H)$. We
also characterize the power bounded property for both $A(H)$ and $B(H).$
