# An Asymptotic Approach in Mahler's Method

### New Zealand Journal of Mathematics

Vol. 47, (2017), Pages 27-42

Michael Coons

School of Math. and Phys. Sciences

University of Newcastle

Callaghan

Australia

Abstract We provide a general result for the algebraic independence of Mahler functions by a new method based on asymptotic analysis. As a consequence of our method, these results hold not only over $\mathbb{C}(z)$, but also over $\mathbb{C}(z)(\mathcal{M})$, where $\mathcal{M}$ is the set of meromorphic functions. Several examples and corollaries are given, with special attention to nonnegative regular functions.

Keywords Algebraic independence, Mahler functions, radial asymptotics

Classification (MSC2000) Primary 11J85; Secondary 11J91, 30B30