Let $\mathrm{erf}(x) := \frac{2}{\sqrt{\pi}} \int_{-\infty}^x \exp(-t^2) \, dt$ be the error function $\mathrm{erf}: \mathbb{R} \to (-1,1)$. It is monotonously increasing and therefore has an inverse $\mathrm{erf}^{-1}: (-1,1) \to \mathbb{R}$.

Now I have two questions:

1) On https://en.wikipedia.org/wiki/Error_function#Inverse_functions it is said that $\mathrm{erf}^{-1}$ can be extended to the open unit disc. Unfortunately, there is no reference given for this. Anyone knows a reference for the derivation of the MacLaurin series of $\mathrm{erf}^{-1}$?

2) There are so-called Hardy spaces $H^p$ of functions analytic on the open disc which satisfy $$\sup_{0<r<1}\left(\frac{1}{2\pi} \int_0^{2\pi}\left|f \left (re^{i\theta}\right )\right|^p \; \mathrm{d}\theta\right)^\frac{1}{p}<\infty.$$ see https://en.wikipedia.org/wiki/Hardy_space#Hardy_spaces_for_the_unit_disk.

I wonder wether $\mathrm{erf}^{-1}$ belongs to some $H^p$ with $p<\infty$?

As it was pointed out on math.SE, where I asked this question before (https://math.stackexchange.com/questions/1546570/inverse-error-function-its-analytic-continuation-and-hardy-space), the MacLaurin coefficients seem to be in $\ell^2$ (which implies membership in $H^2$), but can this also be proven?

Thanks!