Rank-Wise Approximations of Hyper-Operations

[Catullus]

When a^^x converges and a does not equal one, you could possibly extend tetration by using the limit formula here: https://math.eretrandre.org/tetrationfor...42#pid4442. Then you could use the same formula to extend pentation, hexation, et cetera for those bases. Does anyone know of any approximations of those hyper operators that become more and more accurate as the rank of the hyper operation increases?

Hey!

So this formula produces the standard Schroder iteration, or the standard Abel iteration.

I don't think this would work for \(a>\eta=e^{1/e}\), if it does, it would probably look like Kouznetsov's approach to Kneser. But I can 90% guarantee it wont work on the real line.

I can prove this is the standard Schroder/Abel iteration if you like.

For \(a > \eta\) you would have to focus about a fixed point \(L \in \mathbb{C}\), and then you are just performing Schroder about \(L\). This would not produce a real valued tetration about \(a=e\).

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So, technically you are correct. This formula would produce tetration! And it would produce it pretty much everywhere! But, it's the bottom of the barrel for repelling cases, it produces non real values on the real positives. We can't have e^^1/2 being a non-real number...

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That is a cool as **** formula though, lol. Never noticed it before.

I said tetrations of a would converge. a would not be e.
Also why must e^^1/2 be a real number? The Bennet interpolation of the Fibonacci numbers gives a non real number for the halfth Fibonacci number. The tetration being non real valued seems much cooler!
Can you please tell me the proof that this is the standard Schröder/Abel iteration. Also the Abel iteration gives a real number for e^^1/2.