pentation and hexation

[sheldonison]

I'm waiting for an inductive proof that gets you from \( e \uparrow^n x \) to \( e \uparrow^{n+1} x \).

I've been playing around with these functions, but work and life have been busy.  I got hexation with a theta mapping working, which is pretty cool all by itself.   Its actually the inverse of hexation, with roughly 30 decimal digits accuracy for real bases.   Here's a quick picture showing the algorithm, where we sample n points around a circle between the pentation conjugate fixed points; -2.25975437729486 +/- 1.38442438404148*I, and solve for the hexation Abel function on that circle.  On the left, a theta mapping with 60 samples, and 8 theta samples gives 11 decimal digits of precision, with precision of around 30 decimal digits with 240 samples.  On the right, without a theta mapping we are required to extend the sample points all the way to the fixed points.  With 256 samples, we only get 7 decimal digits of precision.  So a theta mapping is pretty important.  The yellow sample points are pairing z with pent(z), and the brown sample points pair z with pent^{-1}(z).  The pink and green points on the bigger circle involve the theta mapping and the Schroder function from the complex fixed point, generated using the 8 green and pink samples in the smaller circle.  Using the sample points in the picture, we setup a linear systems of equations and solve it, much like the routine I use for the slog matrix solution in the latest version of fatou.gp

   

Then we can show this solution between these two conjugate fixed points is unique.  Of course, there may be other fixed points for pentation depending on the base.  Which sort of brings me to a sort of obvious observation.  This pentation and hexation function doesn't seem particularly unique.  There are other real fixed points for pentation for some bases, and there are other complex conjugate fixed points too.  It turns out tetration has complex conjugate fixed points too, so maybe we could have a used a theta mapping to generate an alternative pentation solution as well.  So, we have defined a family of functions that is not particularly unique, but allows analytic superfunctions to at least septation and probably octation. 
(1) pentation defined using leftmost real fixed point of tetration; 
(2) hexation defined using the leftmost complex conjugate fixed points of pentation from (1)
(3) heptation is defined using the leftmost real fixed point of hexation.
(4) ocation defined using the leftmost complex conjugate fixed points of heptation.
.....

I've generated the first pentation, hexation, and heptation, accurate to 25-30 decimal digits.  I've also generated the complex conjugate fixed points for heptation as well, which is often enough to show I can generate a nicely converging theta mapping.  So octation also appears plausible; and octation would have real valued fixed points...  Here is a graph of base(e) exp, sexp, pentation, hexation, and septation!  Octation would probably behave somewhat like hexation.  The pattern is pretty cool, with pentation and heptation tending towards their real valued fixed points as real(z) gets more negative, whereas tetration and hexation have a logarithmic singularity.
   

Let us assume that there is an infinite sequence of these functions.  Now, for base 2, its easy to show that the sequence gets arbitrarily large since the Knuth uparrow function is well defined for integer valued bases.  But Knuth's uparrow only gives us integer values for integer bases and tells us nothing about real valued bases.

So lets make some observation.
Tetration or iterated exponentiation is bounded for bases<=exp(1/e), since there is a real valued upper fixed point.
For pentation Nuinho observed that for bases<= ~1.635 sexp(z) has an upper fixed point, and therefore pentation is bounded for these bases.
For hexation, I observe that for bases<= ~1.738, pent(z) has an upper fixed point, and therefore hexation is bounded for these bases
For heptation, I observe that for bases<=1.799, hex(z) has an upper fixed point, and therefore heptation is bounded for these bases
For octation, I observe that for bases<=1.839, hept(z) has an upper fixed point, and therefore octation is bounded for these bases, assuming one could generation analytic octation ...

Perhaps for all bases<2, the iterated function sequence eventually becomes bounded?