On to C^\infty--and attempts at C^\infty hyper-operations
(03/01/2021, 11:22 PM)MphLee Wrote: 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 inverse Abel-like function of ?

In a strict sense, I don't see how is a Schroeder function of or of .

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 function and then it turns out the function we're iterating is actually
This is the function James is actually iterating when he generates
Schroeder function and inverse formal definition using f(x)

  this works for n=2,3,4 ....

The FPS (formal power series) approach is another intriguing approach to understanding , and the iterated 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 is entire, f has singularities where the derivative of is equal to zero.  Here is the Taylor series for f; the function we are actually iterating to generate , which has a fixed point of
 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 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 also have similar formal series representations, which I have also generated.
+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 
+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
- Sheldon
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 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,

Not much really changes.  I chose to use the exponential convergents rather than the convergents, simply because it generalizes well to the construction of arbitrary super-functions. All we need really is an exponential convergents. In fact, 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 , I figure holomorphy isn't really needed.

I do believe we could definitely use to construct . I believe it's more of an aesthetic issue, and I find it a bit more natural to just use,

And do away with . Also because this satisfies the more natural equation,

And it removes a bit of the untangling if we were to use .

Anyway, here's what I have so far. The proof of hyper-operators is surprisingly copy/paste from the proof of 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

Attached Files
.pdf   Hyper_operators.pdf (Size: 338.85 KB / Downloads: 197)

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