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 On to C^\infty--and attempts at C^\infty hyper-operations sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 03/02/2021, 05:09 AM (This post was last modified: 03/02/2021, 10:32 PM by sheldonison.) (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 $\phi_{n+1}$ 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 - Sheldon JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 03/02/2021, 09:55 PM 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 Attached Files   Hyper_operators.pdf (Size: 338.85 KB / Downloads: 100) « Next Oldest | Next Newest »

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