Galois theories for $q$-difference equations: comparison theorems

The content of this paper was previously included in arXiv:1002.4839We establish some comparison results among the different Galois theories, parameterized or not, for q-difference equations, completing the work of Chatzidakis-Hardouin-Singer. Our main result is the link between the abstract parameterized Galois theories, that give information on the differential properties of abstract solutions of $q$-difference equations, and the properties of meromorphic solutions of such equations. Notice that a linear q-difference equation with meromorphic coefficients always admits a basis of meromorphic solutions