Superroots and a generalization for the Lambert-W

[Gottfried]

Conjecture 3.1 fails because the left hand side has radius going to 0.

Hmm, I've not yet settled everything about this in my mind. I've of course seen, that with increasing n the convergence-radius of the function \( \;^n W \) decreases. However, as usually, if a function can be analytically continued (beyond its radius of convergence) for instance by Euler-summation, I assume, that the result is still meaningful. And we have here the possibility for Euler-summation, so I think there is a true analytic continuation. However, I don't know yet whether this can be correctly inserted in my conjecture-formula for the limit-case.

It is a mystery how you intend to solve the cases x^^(3/2) = v though.

As I understand this, this is using fractional iteration heights. As I described my exercises, I'm concerned with the unknown bases, and am using integer heights so far, not fractional heights ( superroot, not superlog)

Gottfried