  For lattice polytopes $P_1,\ldots, P_k \subseteq \mathbb{R}^d$, Bihan (2014)
introduced the discrete mixed volume $\mathrm{DMV}(P_1,\dots,P_k)$ in analogy
to the classical mixed volume. In this note we initiate the study of the
associated mixed Ehrhart polynomial $\mathrm{ME}_{P_1,\dots,P_k}(n) =
\mathrm{DMV}(nP_1,\dots,nP_k)$. We study properties of this polynomial and we
give interpretations for some of its coefficients in terms of (discrete) mixed
volumes. Bihan (2014) showed that the discrete mixed volume is always
non-negative. Our investigations yield simpler proofs for certain special
cases. We also introduce and study the associated mixed $h^*$-vector. We show
that for large enough dilates $r P_1, \ldots, rP_k$ the corresponding mixed
$h^*$-polynomial has only real roots and as a consequence the mixed
$h^*$-vector becomes non-negative.
