We study parametrized linear differential equations with coefficients depending
meromorphically upon the parameters. As a main result, analogously to the
unparametrized density theorem of Ramis, we show that the parametrized
monodromy, the parametrized exponential torus and the parametrized Stokes
operators are topological generators in the Kolchin topology for the parametrized
differential Galois group introduced by Cassidy and Singer. We prove an analogous
result for the global parametrized differential Galois group, which generalizes a result
by Mitschi and Singer. These authors give also a necessary condition on a group for
being a global parametrized differential Galois group; as a corollary of the density
theorem, we prove that their condition is also sufficient. As an application, we give a
characterization of completely integrable equations, and we give a partial answer to a
question of Sibuya about the transcendence properties of a given Stokes matrix.
Moreover, using a parametrized Hukuhara–Turrittin theorem, we show that the
Galois group descends to a smaller field, whose field of constants is not differentially
closed.
