Steenrod Algebra and lacunary power series

The dual Steenrod algebras are supercommutative Hopf algebras, so their spectra are algebra supergroup schemes. These group schemes are closely related to the automorphisms of 1-dimensional additive formal groups. For example, if ”p”=2 then the dual Steenrod algebra is the group scheme of automorphisms of the 1-dimensional additive formal group scheme ”x”+”y” that are the identity to first order. These automorphisms are of the form
x\rightarrow x + \xi_{1}x^{2}+\xi_{2}x^{4}+\xi_{3}x^{8}+\cdots

To an analyst, that power series is beguilingly close to the function
F(q) = \sum_{n=1}^{\infty} q^{2^{n}}

What can Steenrod algebra tell us about such series, and contrariwise, what can the analysis of such series tell us about Steenrood algebras?

Which can be read about in This paper by Ahmed Sebbar, and This m.se post. Also of note,
The Many Faces of the Kempner number

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