Statement about our current stage of findings?

[bo198214]

So, why not consider a statement, which qualifies the now achieved collection of all these results?

We have (at least) 3 different analytic extensions for tetration and no(t one!) proof whether any two of these extensions are equal or different (where they overlap).

The extensions methods are:

1. Natural Abel function: \( b>1 \) (Andrew, is this restriction necessary?) applicable to arbitrary non-fixed points.
   
2. Regular iteration at the lower fixed point: \( e^{-e}<b<e^{1/e} \), regular iteration is applicable only at fixed points.
   
3. Diagonalization method: bases?, applicable to fixed points and non-fixed points.
   
4. Jay's method: bases?
   

We have only the particular result that the diagonalization method at a fixed point is equal to regular iteration.

Second: what are the open problems?

H1: Are the extensions achieved by those methods equal?

A first simpler question in that direction would be whether the natural Abel method applied to multiplication gives indeed exponentiation, or more specifically whether the natural Abel function of \( ex \) developed at 1 indeed is \( \ln(x) \). While we already know that the regular/diagonalization tetration of \( ax \) developed at 0 is \( a^x \).

H2: Proof for convergence of the coefficients for the natural Abel series. Proof of convergence (radius) for the natural Abel series.
H3: Same for the diagonalization method.

2) nonuniqueness wrt shifting at different fixpoints, when non-real-integer heights are involved

My question would first be whether the natural Abel method yields the same result regardless of the development point.
Same question for the diagonalization method.

1) As far as we use basically powerseries-representation for the tetration/decremented exponentiation: divergent series occur with non-real-integer heights

For example?

4) Extensions to higher (or zero-) order hyperoperators

Once we solved the problem of equality of the methods, the (first three) methods can easily applied to any hyper operation giving the super hyper operation ( ).