Let $\left\{{R}_{n}\right\}$ and $\left\{{P}_{n}\right\}$ be two polynomial systems which induce signed polynomial hypergroup structures on ${N}_{0}$. The paper under review investigates when the Banach algebra ${l}^{1}({N}_{0},{h}^{R})$ can be continuously embedded into or is isomorphic to ${l}^{1}({N}_{0},{h}^{P})$. Certain sufficient conditions on the connection coefficients ${c}_{n,k}$ given by ${R}_{n}={\sum}_{k=0}^{n}{c}_{nk}{P}_{k}$, for the existence of such an embedding or isomorphism are given. These results are also applied to obtain amenability properties of the ${l}^{1}$-algebras induced by Bernstein-Szegő and Jacobi polynomials.

The previous related investigations can be found in *W. R. Bloom* and *M. E. Walter*’s work [J. Aust. Math. Soc., Ser. A 52, No. 3, 383–400 (1992; Zbl 0776.43001)], which was only concerned with the isometric isomorphisms of hypergroups. For more recent works, see *R. Lasser*’s articles [Stud. Math. 182, No. 2, 183–196 (2007; Zbl 1126.43003); Colloq. Math. 116, No. 1, 15–30 (2009; Zbl 1167.43007)], which studied the amenability of ${l}^{1}$-algebras of polynomial hypergroups.