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*

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