  For an $n$-dimensional lattice simplex $\Delta_{(1,\mathbf{q})}$ with
vertices given by the standard basis vectors and $-\mathbf{q}$ where
$\mathbf{q}$ has positive entries, we investigate when the Ehrhart
$h^*$-polynomial for $\Delta_{(1,\mathbf{q})}$ factors as a product of
geometric series in powers of $z$. Our motivation is a theorem of
Rodriguez-Villegas implying that when the $h^*$-polynomial of a lattice
polytope $P$ has all roots on the unit circle, then the Ehrhart polynomial of
$P$ has positive coefficients. We focus on those $\Delta_{(1,\mathbf{q})}$ for
which $\mathbf{q}$ has only two or three distinct entries, providing both
theoretical results and conjectures/questions motivated by experimental
evidence.
