Hypertranscendence of solutions of Mahler equations

Abstract

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 $u_1,\dots ,u_n$ being given, are $u_1,\dots ,u_n$ 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., these series and their successive derivatives are algebraically independent. Our approach relies on 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).