Marius van der Put (Groningen): Hyper transcendental functions

An analytic function f is called transcendental if a_n f^n + ... + a_1 f+a_0=0, with all a_i in C(z) (rational functions in z over the complex numbers), implies that all a_i=0. Some analytic functions are more transcendental than others. These are called hyper transcendental. An analytic function f has this property if the collection {f^(m) | m >= 0} of all derivatives of f is algebraically independent over C(z). The Gamma function is hyper transcendental. We will develop some differential Galois theory and some difference Galois theory in order to analyze hyper transcendence. Finally, we will discuss a recent theorem of Ch. Hardouin.