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

[sheldonison]

\( \phi_3(s)=\phi_{2}(\phi_{3}(s-1)+s) \) probably grows more like pentation.
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So sorry but .. I am not convinced of its usefullness.
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It does not seem simpler than generalizing ackermann , making ackermann analytic etc 

Regards 

tommy1729

Hey Tommy,

I mostly agree with you with one small caveat. Here is my observation.

Kneser's tetration function is actually beautifully behaved in the complex plane, or at least as reasonably nicely behaved as any tetration superfunction can be.  The same thing can be said for Jame's \( \phi_2 \) function, except that it is also entire and 2pi*i periodic.

Now, all of the "analytic" higher order functions like pentation generated from the lower fixed point of Kneser's Tetration are actually pretty poorly behaved in the complex plane.  All those singularities at the negative integers for tetration get reflected in pentation as real(z) grows and they eventually show up arbitrarily close to the real axis.  I have experimented with an analytic hexation generated from a complex conjugate pair of fixed points from such an "analytic" pentation, but it too is really poorly behaved in the complex plane.  So imho, the higher order analytic Ackermann functions after tetration don't seem as interesting to me as tetration.

Contrast this with the family of \( \phi_n \) functions all of which seem to be entire and 2pi*i periodic. 

Since they are 2pi*I periodic with limiting behavior as real(z) grows arbitrarily negative of \( \exp(z) \), then each of these functions has a corresponding Schroeder function whose inverse \( \Psi^{-1} \) is also entire with
\( \phi_n(z)=\Psi_n^{-1}(\exp(z)) \)
\( \Psi^{-1}_n(x)=x+\sum_{n=2}^{\infty}a_n\cdot x^n;\;\; \) there is a formal entire inverse Schroeder function for each phi_n function.
Moreover, each formal Schroeder function's Taylor series can be generated with surprising ease.  For \( \Psi^{-1}_2(x)=x+... \) we initialize a function f=x, and we iterate n times to calculate n+1 Taylor series terms as follows, where each iteration gives one additional exact term in the Taylor series of the inverse Schroeder function.
\( f_1=x; f_n(x)=x\cdot\exp(f_{n-1}(\frac{x}{e}));\;\;\Psi^{-1}_2(x)=\lim_{n\to\infty}f_n(x);\;\;\Psi^{-1}_2(x)=x+\frac{x^2}{e}+... \) 

Surprisingly, the iteration for \( \Psi^{-1}_3(x);\Psi^{-1}_4(x);\Psi^{-1}_5(x);\; \) are only a little bit more complicated but can also easily be coded in a single lines of pari-gp code or mathematical equations as follows:

\( g_1=x; g_n(x)=\Psi^{-1}_2(x\cdot\exp(g_{n-1}(\frac{x}{e})));\;\;\Psi^{-1}_3(x)=\lim_{n\to\infty}g_n(x);\;\; \)this works for Psi_4,5 etc.

So I would assert that this family of iterated entire superfunctions is more well behaved than any other family of iterated superfunctions that I am aware of.  It is also more accessible  and easy to calculate than any other family than I am aware.   In that sense, it is also more accessible than Kneser, which is actually pretty difficult to understand and calculate.