  The last years have seen a growing interest from mathematicians in Mahler
functions. This class of functions includes the generating series of the
automatic sequences. The present paper is concerned with the following problem,
which is omnipresent in combinatorics: a set of Mahler functions
$u_{1},...,u_{n}$ being given, are $u_{1},...,u_{n}$ and their successive
derivatives algebraically independent? In this paper, we give general criteria
ensuring an affirmative answer to this question. We apply our main results to
the generating series attached to the so-called Baum-Sweet and Rudin-Shapiro
automatic sequences. In particular, we show that these series are
hyperalgebraically independent, i.e., that these series and their successive
derivatives are algebraically independent. Our approach relies of the
parametrized difference Galois theory (in this context, the
algebro-differential relations between the solutions of a given Mahler equation
are reflected by a linear differential algebraic group).
