  We study parameterized linear differential equations with coefficients
depending meromorphically upon the parameters. As a main result, analogously to
the unparameterized density theorem of Ramis, we show that the parameterized
monodromy, the parameterized exponential torus and the parameterized Stokes
operators are topological generators in Kolchin topology, for the parameterized
differential Galois group introduced by Cassidy and Singer. We prove an
analogous result for the global parameterized 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 parameterized 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
parameterized Hukuhara-Turrittin theorem, we show that the Galois group
descends to a smaller field, whose field of constants is not differentially
closed.

Classical Analysis and ODEs