I wrote some stuff on power series related to fractional iteration of functions (like solving f(f(x))=sin x) http://blog.sigfpe.com/2017/02/logarithms-and-exponentials-of-functions.html …
I figured out the case derived from cos(2^n arccos(x)) long ago but couldn't progress further. Surely there's a generalization.
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There won't be progress if progress isn't possible. I suspect that there is no useful generalisation.
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