  This work merges two different types of many-electron ensembles, namely the
Theophilou-Gross-Oliveira-Kohn ensembles of ground and neutrally-excited
states, and the more recent $N$-centered ensembles of neutral and charged
ground states. On that basis, an in-principle exact and general, so-called
extended $N$-centered, ensemble density-functional theory of charged and
neutral electronic excitations is derived. We revisit in this context the
concept of density-functional derivative discontinuity for neutral excitations,
without ever invoking nor using the asymptotic behavior of the ensemble
electronic density. The present mathematical construction fully relies on the
weight dependence of the ensemble Hartree-exchange-correlation
density-functional energy, which makes the theory applicable to lattice models
and opens new perspectives for the description of gaps in mesoscopic systems.
