In the present paper, we use difference Galois theory to study the nature of the generating series counting walks in the quarter plane. These series are trivariate formal power series Q(x, y, t) that count the number of discrete paths confined in the first quadrant of the plane with a fixed directions set. While the variables x and y are associated to the ending point of the path, the variable t encodes its length. In this paper, we prove that if Q(x, y, t) does not satisfy any algebraic differential relations with respect to x or y, it does not satisfy any algebraic differential relations with respect to the parameter t. Combined with [BBMR16, DHRS18, DHRS17], we are able to characterize the t-differential transcendence of the generating series for any unweighted walk.

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 rather frequently encountered in combinatorics: a set of Mahler functions u 1 , ..., un being given, are u 1 , ..., un 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). 14 3.

In the present paper, we use Galois theory of difference equations to study the nature of the generating series of (weighted) walks in the quarter plane with genus zero kernel. Using this approach, we are able to prove that the generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational coefficients.