An equivalent condition for q-holonomicity

We show that a sequence is q-holonomic if and only if it satisfies the elimination property for any subset of variables. The same result also holds for holonomic sequences. As an application, we prove several conjectured closure properties for q-holonomic sequences. We also prove that Jones-style sequences for links in any closed $3$-manifold are q-holonomic, which in turn implies that the Reshetikhin-Turaev invariants are q-holonomic in the colors.