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Ah, but $2^{2^n}$ is a hypergeometric function, right? I guess, I confused hypergeometric function (nearly every function used in standard mathematics) and hypergeometric term ($\frac{f(x+1)}{f(x)}$ is rational)
en.wikipedia.org/wiki/… has an explicit formular for $H_\frac{p}{q}$, though the approximations for cos, ln,... will be annoying, I'll update the question once I implemented it
