Solving tetration for base 0 < b < e^-e

[mike3]

Is there any way in which this could be done with something other than the matrix operator? For example, could a limit formula be used like that for \( b = e^{1/e} \), for this base (\( b = e^{-e} \)) even if convergence is dog slow?

Hm actually, I am not aware of such a limit formula. I only know \( |\lambda|\neq 0,1 \) and \( \lambda=1 \) but there should be also a formula somewhere for your case of \( \lambda=-1 \) (or generally for \( \lambda^m = 1 \) for some \( m \), here \( m=2 \)). If I have some time I will look through some books.

The powerseries development of the regular slog/sexp probably anyway has zero convergence radius.

One can get a limit formula for \( b = e^{1/e} \) as the behavior at large values of the tower is somewhat like \( e - \frac{2e}{x} \) (x = tower). And the bases \( e^{-e} < b < e^{1/e} \) have exponential behavior (exponential-like convergence to their fixed point) at large towers. But what sort of behavior does \( e^{-e} \) have at large values of the tower? It's not exponential and it's also not a simple rational function either (as it was for \( b = e^{1/e} \)).