Holm [Holm 04, Holm 02] studied modules of higher-order differential operators (generalizing derivations) on generic (central) hyperplane arrangements. We use his results to determine the Hubert series of these modules. We also give a conjecture about the Poincaré-Betti series, these are known for the module of derivations through the work of Yuzvinsky [Yuzvinsky 91] and Rose and Terao [Rose and Terao 91 ] © A K Peters, Ltd.