Here we prove an analogue of Theorem 2.1 for the function
which was considered by Mahler. Now this series converges in the complex plane only on the unit disc D defined by |z| < 1; and indeed it has a so-called natural boundary on |z| = 1 which precludes any further analytic continuation. Near z = 0 it converges rather more rapidly than and so we might expect things to be easier here. So let us again see what truncation of f(α) gives.
Let us start with α = a/b, for simplicity taking a ≥ 1, b ≥ 1; but now a