Galois theory of difference equation and differential transcendence

I will explain how the parameterized Galois theory of difference equations gives a systematic approach to differential transcendence. As an example I will give a Galoisian proof of Holder theorem for Euler Gamma functions. These results are contained in a work by C. Hardouin and M. Singer. I will, then, speak of a joint work with C. Hardouin about differential relations of solutions of $q$-difference equations.