Are there any noteworthy developments about tetration in the last six months? Ember Edison Fellow Posts: 92 Threads: 10 Joined: May 2019   02/09/2023, 05:20 AM I didn't resist the effects of COVID-19 as easily as I thought I would. I hope I didn't miss anything interesting. JmsNxn Ultimate Fellow Posts: 1,214 Threads: 126 Joined: Dec 2010 02/10/2023, 03:27 AM (This post was last modified: 02/10/2023, 03:28 AM by JmsNxn.) I think the only real "tetration-y" stuff that has happened here--at least on mine and Gottfried's part has been discussing Neutral fixed points. Where we have stumbled across a bunch of literature which confirms that there is no holomorphic function: $g(g(x)) = \eta^x\\$ For $x$ in a neighborhood of $e$. And we were able to find proofs of very tight bounds on this series; where: $g(x) = e + \sum_{k=1}^\infty a_k (x-e)^k\\$ Where: $a_k = O(c^k k!)$ This led me to write a half finished paper, I should get back to, about iterating around neutral fixed points in general. It didn't really go anywhere though really. But we did cover a lot of ground on asymptotic series expansions near neutral fixed points; and their relation to Abel functions, so that's cool ! The rest of the stuff that's happened here has been the usual whacky "tetration adjacent" stuff, lol Ember Edison Fellow Posts: 92 Threads: 10 Joined: May 2019 02/11/2023, 08:34 PM (02/10/2023, 03:27 AM)JmsNxn Wrote: I think the only real "tetration-y" stuff that has happened here--at least on mine and Gottfried's part has been discussing Neutral fixed points. Where we have stumbled across a bunch of literature which confirms that there is no holomorphic function: $g(g(x)) = \eta^x\\$ For $x$ in a neighborhood of $e$. And we were able to find proofs of very tight bounds on this series; where: $g(x) = e + \sum_{k=1}^\infty a_k (x-e)^k\\$ Where: $a_k = O(c^k k!)$ This led me to write a half finished paper, I should get back to, about iterating around neutral fixed points in general. It didn't really go anywhere though really. But we did cover a lot of ground on asymptotic series expansions near neutral fixed points; and their relation to Abel functions, so that's cool ! The rest of the stuff that's happened here has been the usual whacky "tetration adjacent" stuff, lol Fortunately, I really didn't miss much. And congratulations on your discovery.  The neutral fixed points is quite mysterious, and it's a good thing to have more understanding. « Next Oldest | Next Newest »

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