An asymptotic approach in Mahler's method

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 all meromorphic functions. Several examples and corollaries are given, with special attention to nonnegative regular functions.