(06/08/2017, 01:19 PM)sheldonison Wrote: Tetration has real valued fixed points, so you don't need another Kneser Mapping for pentation, but such pentation functions don't seem all that interesting. Personally, Kneser's Tetration holds a special place, and seems much more fun than the bounded Tetration solutions for bases<=eta, and also more interesting then pentation.

Really? Does the same happen for pentation? Does it have a real fixed point? Has anybody bothered to go around and actually construct using Kneser's tetration as a base? I mean, if tetration has a repelling real fixed point then it's easy to get to pentation. If pentation has the same thing, it's easy to get hexation. So on and so forth. Though of course, computationally it'd probably be exhausting and impractical. I mean, it would probably be the most taxing thing on a computer known to man to compute something like . But if they are well behaved a nice proof by induction may work. That's all you need to get . That to me is the holy grail of mathematics, constructing .

I'd also like to add that the bounded case has some pretty insane properties. I think it's special in a different sense compared to our usual tetration. It's just as interesting. It is bounded. It is exponentially decaying. It is periodic. It has a Fourier series expansion.

for some Dirichlet series . Which is pretty cool. They're also Newton summable. They're totally monotone (as we've so recently uncovered for tetration, and I've recently come to believe for any arbitrary bounded analytic hyper operator).

All in all they do some pretty crazy stuff.