An explanation for this?

[bo198214]

So would this mean that Andrew's algorithm and Kneser's construction yield the same result?

Check out Jayd's findings yourself here, it seems numerically giving the same results.

At the very least, it seems to be pointing at the idea that the tetration is a kind of function that really has no precedent, an exotic, "alien" kind of function that is really a whole lot different from the conventionally accepted set of special functions. Is this a fair assessment?

Well the discussing of the (existence and uniqueness) of perturbed Fatou coordinates (i.e. Abel function at a fixed point pair) is relatively recent in holomorphic dynamics. I think it was not discovered before *because* we have no nice coefficient description. Its a different/non-classic way which enables to tackle the problem.

no closed form coefficients ? Riemann mapping theorem states there is a holomorphic function !?

so we have a taylor series with its coefficients.

and although not trivial , those are computable not ?

maybe but thats not called a closed form if you can somehow compute it.