On some arithmetic properties of Mahler functions

Mahler functions are power series f(x) with complex coefficients for which there exist a natural number n and an integer ℓ ≥ 2 such that f(x), f(xℓ),..., f(xℓn−1),f(xℓn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({x^{{\ell ^{n - 1}}}}),f({x^{{\ell ^n}}})$$\end{document} are linearly dependent over ℂ(x). The study of the transcendence of their values at algebraic points was initiated by Mahler around the’ 30s and then developed by many authors. This paper is concerned with some arithmetic aspects of these functions. In particular, if f(x) satisfies f(x) = p(x)f(xℓ) with p(x) a polynomial with integer coefficients, we show how the behaviour of f(x) mirrors on the polynomial p(x). We also prove some general results on Mahler functions in analogy with G-functions and E-functions.