"Natural boundary", regular tetration, and Abel matrix

[bo198214]

Hmm. I dug up this:

http://www.math.wustl.edu/~sk/limits.pdf

It says that if a function converges pointwise in a region, the convergent need only be holomorphic on a dense, open subset.

Oh thats an interesting one, I didnt yet hear of it (I only know the compact convergence holomorphy). Really interesting.

Note that dense can *not* imagined here as something like \( \mathbb{Q} \) is dense in \( \mathbb{R} \), because a function holomorphic in a point, implies the function being holomorphic in some neighborhood! Here it rather can be considered as that the set of points where the limit is not holomorphic does not contain inner points (i.e. there can be no small disk where the limit is not holomorphic).
This is really a strong result!

However, as you wrote, in our case there may well be a boundary where the limit is not holomorphic (and that may even be different from the STB if the intuitive iteration differs from regular iteration).

Failing that, could examining the derivatives on the part of the STB where it converges be useful? If the convergent is not holomorphic on the STB, then wouldn't these derivatives explode, go nuts, etc.?

One would have to show that \( \limsup_{n\to\infty}\sqrt[n]{\left|a_n\right|}=\infty \), i.e. that there is some subsequence of coefficients that goes to infinity \( \sqrt[n_k]{\left|a_{n_k}\right|} \to \infty \).

But actually the intuitive iteration is really difficult to handle, that's why there are nearly no results about it. No convergence proof, not even a proof that \( \log_b \) is the intuitive Abel function of \( bx \) (which is a really simple function compared to \( \exp_b \) which is the interest for tetration).