On to C^\infty--and attempts at C^\infty hyper-operations

[sheldonison]

I'm probably missing some key piece of the puzzle (terminology). Are you talking about a kind of inverse Schroeder-like function right? A confortable abuse of name similar to how we can call \( \phi_{n+1} \) inverse Abel-like function of \( \phi_{n} \)?

In a strict sense, I don't see how \( \psi_{n} \) is a Schroeder function of \( \phi_{n} \) or of \( \phi_{n-1} \).

Sorry for the confusion; yes you are correct.  The Schroeder like function is Schroeder like only in that it has a formal power series beginning with x+a2x^2 ... and a multiplier at zero, with the multiplier=e.  Since it is a formal series, we can get the formal inverse and generate a \( \Psi(x) \) function and then it turns out the function we're iterating is actually
\( f(x)=e^{(x+1)}\cdot\Psi(x);\;\; \) This is the function James is actually iterating when he generates \( \phi \)
\( \Psi(f(x))=e\cdot\Psi(x);\;\;\; \) Schroeder function and inverse formal definition using f(x)
\( \Psi^{-1}(e\cdot~x)=f(\Psi^{-1}(x)) \)
 \( \phi(x)=\Psi^{-1}(e^x);\;\;\;\phi_n(x)=\Psi_n^{-1}(e^x);\;\; \) this works for n=2,3,4 ....

The FPS (formal power series) approach is another intriguing approach to understanding \( \phi \), and the iterated \( \phi_n \) functions.  The FPS approach would need more effort to make it rigorous; and the effort to make the FPS rigorous might become increasingly daunting for the iterated phi series for n>2.  Even though \( \phi \) is entire, f has singularities where the derivative of \( \Psi^{-1} \) is equal to zero.  Here is the Taylor series for f; the function we are actually iterating to generate \( \phi \), which has a fixed point of \( f(0)=0;\;f^{'}(0)=e \)

Code:

{f=
 x^ 1*  2.71828182845905
+x^ 2*  1.71828182845905
+x^ 3*  0.775624792750073
+x^ 4*  0.191889268327428
+x^ 5*  0.0520249429156080
+x^ 6*  0.00599242247026314
+x^ 7*  0.00182349994116415
+x^ 8*  9.81807721872041 E-5
+x^ 9* -5.19018256906951 E-5
+x^10*  7.84647429007181 E-5
+x^11* -5.26096655278693 E-5
+x^12*  3.02110037576056 E-5
+x^13* -1.51896837385654 E-5
+x^14*  6.77204817742090 E-6
+x^15* -2.59325526178607 E-6 ...

Here are the first few Taylor series coefficients of the \( \Psi^{-1}(x) \) function which is entire.  We can generate the individual terms with a closed form in terms of "e", but I don't have a generic equation for the closed form.  The higher order pentation, and hexation \( \Psi_3^{-1};\;\Psi_4^{-1} \) also have similar formal series representations, which I have also generated.

Code:

x
+x^ 2*  0.367879441171442
+x^ 3*  0.117454709986170
+x^ 4*  0.0324612092929206
+x^ 5*  0.00811730704942829
+x^ 6*  0.00188547471967479
+x^ 7*  0.000413224905195451
+x^ 8*  8.63482541739982 E-5
+x^ 9*  1.73333585608164 E-5
+x^10*  3.36137276288664 E-6
+x^11*  6.32477784106711 E-7
+x^12*  1.15869533017107 E-7
+x^13*  2.07255547935482 E-8
+x^14*  3.62795192280962 E-9
+x^15*  6.22699185709248 E-10 + ...

Finally, for completeness here are the first few Taylor series terms of the formal series for \( \Psi(x) \)

Code:

x
+x^ 2* -0.367879441171442
+x^ 3*  0.153215856487055
+x^ 4* -0.0653506857689096
+x^ 5*  0.0275282379807258
+x^ 6* -0.0111894054323465
+x^ 7*  0.00428067464337933
+x^ 8* -0.00147921549686095
+x^ 9*  0.000417655162777504
+x^10* -5.90712473237761 E-5
+x^11* -3.60198809495273 E-5
+x^12*  4.43758017488974 E-5
+x^13* -3.14471384470661 E-5
+x^14*  1.81562489170478 E-5
+x^15* -9.15769831156020 E-6

[JmsNxn]

Hey Everyone. Haven't been on for a while, been a little busy.  I thought I'd post the modified form of the paper.

It details how to construct \( C^\infty \) hyper-operations; minus maybe a few details. But I'm confident I got everything down. In fact, I'm confident that if we do it in the manner Sheldon describes, using a sequence of infinite compositions,

\(
\phi_n(s) = \Omega_{j=1}^\infty \phi_{n-1}(s-j+z)\bullet z\\
\)


Not much really changes.  I chose to use the exponential convergents rather than the \( \phi_n \) convergents, simply because it generalizes well to the construction of arbitrary super-functions. All we need really is an exponential convergents. In fact, \( \phi_n \) will look pretty much similarly, because of its exponential nature. The better form of Sheldon's method is that we retain holomorphy. Since I'm only trying to show \( C^\infty \), I figure holomorphy isn't really needed.

I do believe we could definitely use \( \phi_n(s) \) to construct \( e \uparrow^n x \). I believe it's more of an aesthetic issue, and I find it a bit more natural to just use,

\(
\Phi_n(s) = \Omega_{j=1}^\infty e^{s-j}e \uparrow^{n-1} z\bullet z\\
\)

And do away with \( \phi \). Also because this satisfies the more natural equation,

\(
\Phi_n(s+1) = e^s e \uparrow^{n-1} \Phi_n(s)\\
\)

And it removes a bit of the untangling if we were to use \( \phi_n \).

Anyway, here's what I have so far. The proof of \( C^\infty \) hyper-operators is surprisingly copy/paste from the proof of \( C^\infty \) tetration, so long as you pay attention to the generalization, it should be fine.

Any questions, comments or the what have you are greatly appreciated. I spent a lot of time restructuring the paper, but a lot of it is similar to what I wrote before.

Thanks, James