  Starting from a complete family (not defined by the reproducing kernel) for
the unit sphere $\mathbf S^n$ in the complex $n$-space $\mathbb C^n$, we obtain
an asymptotic expansion for the associated Berezin transform. The proof
involves the computation of the asymptotic behaviour of functions in the
complete family. Furthermore, we prove an Egorov-type theorem for the covariant
symbol related to a pseudo-differential operator on $L^2(\mathbf S^n)$.
