  We discuss the reflexivity of hyperexpansions and their Cauchy dual
operators. In particular, we show that any cyclic completely hyperexpansive
operator is reflexive. We also establish the reflexivity of the Cauchy dual of
an arbitrary $2$-hyperexpansive operator. As a consequence, we deduce the
reflexivity of the so-called Bergman-type operator, that is, a left-invertible
operator $T$ satisfying the inequality $TT^* + (T^*T)^{-1} \leqslant 2
I_{\mathcal H}.$
