Statement about our current stage of findings?

[Gottfried]

I'm asking myself, how far we are, to assume the tetration-problem to be solved.

We have pretty much achieved with different bases, with complex heights, and different initial parameter in the sense

or in our shorter current notation, where x =1 is assumed


or the "decremented"- variant of this.

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


Second: what are the open problems?

In my view (surely biased by my own involvement) it is
[update]
1) As far as we use basically powerseries-representation for the tetration/decremented exponentiation: divergent series occur with non-integer heights (even if only real and not additionally complex) [/update]

2) nonuniqueness wrt shifting at different fixpoints, when non-integer heights are involved (see [update]-remark)

3) infinite series of powertowers (alternating sign, for the time being)
3.1) where consecutive x are involved, b and h are constant and natural numbers
3.2) where consecutive heights are involved, x and b are constant, b in the range of convergence of the infinite powertower - or even more general
3.3) where consecutive b are involved, x and h are constant, h natural

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

--------------------------------------------------

For 1) the well established Euler-summation is not always sufficient: what method of summation of higher order can be applied to assign values to such powerseries, which are then still consistent with applications of arbitrary further common algebraic manipulations?


For 2) the difference of results, when shifted at different fixpoints, makes the definition non-unique. But how are these different results related? Possibly in a sinusoidal relation, like the zeta-function of positive and negative argument, which are related by a cos()-factor.
Can we determine this relation?


3) I've only discussed alternating series so far since we have then a possibility to check the matrix-results against conventional summation. A functional relation with the non-alternating series, as it can nicely be done with the zeta()/eta()-functions seems out of reach yet.

3.1) This seems to be the most simple one; the crosschecking with serial summation confirm the findings by the matrix-method in applicable ranges of parameters, so at least the systematics of the findings and then the evolving generalizations should be repeated as conjectures.

3.2) The crosscheck by serial summation show, that the two-way-infinite series (or sum of the two one-way-infinite series) by matrix-method shows an effect of error, which possibly can be qualitatively described when its relation to laurent-series is considered. Indeed, using one-way-infinite series with arbitrary finite start-index "on the other side", so h=-j to inf (where h is the index and thus the height) with a finite j seems to agree with the sums, computed by conventional methods in the applicable range of parameters, and show smooth extension beyond these ranges.
The differences, when matrix-method and serial-summation are compared with two-way-infinite extension seem again being sinusoidal with an amplitude depending on the base, so we may have a chance to quantify this difference and establish a proper functional relation between h=0..inf and h=-inf..0 introducing a cos()-factor and some base-depending scaling.
Given, that at least the one-way-infinite series agree with the conventional summation for applicable ranges of parameters, and the continuation using the matrix-methods looks smooth, we should consider to restate a conjecture, after its extent of range of possible validity is more intensely checked.

3.3) Little is done here; only the sums can be expressed by sums of powers of logs(b), where b are the consecutive bases. The sums-of-like-powers of logs are Euler-summable, and the matrix-method gives then diverging sums of these log-sums, which may be discussed further and their relation may be found interesting sometime.



4) Here we are still in speculation, having found some nice individual results for limit cases, and/or in the process of finding a common sense for the definitions, for instance the definition of zeration.

-----------------------------------

Well, these are some "open problems" from my view, surely being not aware of all subjects and proceedings we had in our 6/7-month exchange here. My main impulse is to trigger us to step aside a bit and try to sketch our results in a whole (while momentary) picture, which can then be transmitted to the mathematical community - if you share my opinion, that we indeed have settled something here.

Gottfried
[update] The term "non-real-integer" was meant to express things with most precision, but it may itelf be misleading. I wanted to say
"when non-integer , even if only real and not (additionally) complex"

[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: (Andrew, is this restriction necessary?) applicable to arbitrary non-fixed points.
   
2. Regular iteration at the lower fixed point: , 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 developed at 1 indeed is . While we already know that the regular/diagonalization tetration of developed at 0 is .

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 ( ).

[Gottfried]

So, let us add these remarks to an explicite list of "open problems" which we have discerned here (or know by literature).

My intention is to produce such an overview - possibly also to help ourselves to (re-) focus on the basic and important subjects of research.

[andydude]

It sounds like we should be producing 2 documents. The "FAQ", and the "Reference". I believe that the "Open Problems" section should be listed in the "Reference" document, and that this should be our guide as to what problems remain unsolved. There could also be a section for "Closed Problems" to remind people what was recently solved (of course, with pointers or references to places in the document where the solution can be found). I also think that these 2 documents have been delayed in part because of a lack of structure in collaboration. For a while I was under the impression that we were to produce a single FAQ for everything, but then when I produced my version of the FAQ, there was a lot of reactions to its verbosity, indicating that we should separate those ideas into a FAQ and a Reference document. Another thing I have noticed an inconsistency in filenames. I think in terms of accessibility (to newbies to the forum) all future FAQ and Reference revisions should follow the conventions "tetration-faq-yyyymmdd" and "tetration-ref-yyyymmdd" instead of the past filenames (20070809tetrationfaq, tetration-formula, and FAQ_20080112). This would add a lot of consistency that would be beneficial for people looking for the latest revisions. I will incorporate these filenames into my future revisions.

As for the open questions mentioned in this thread, these are really good questions. I need to re-read many things before I can attack these problems. I am still having difficulty with the regular/matrix approaches. I have read Gottfried's approach many times, and each time I understand a little bit more, but I must admit that I find Aldrovandi's discussion of matrices seem to be much more understandable. What are the differences between Aldovandi's approach and Gottfried's approach? Does "diagonalization" imply the standard diagonalization? or some non-standard diagonalization? Are the diagonalization approach and Gottfried's matrix approach the same? If not, then how are these approaches different from the Carleman matrix applied to the Schroeder functional equation? When applying the Carleman matrix to the Schroeder functional equation, are the diagonals necessarily powers of the fixed point? Correspondingly, do all functions that satisfy the Schroeder functional equation require using a fixed point as a "base" of sorts (defining the Schroeder function as the base-(fixed point) exponential of the Abel function)? Is Aldrovandi's eigen-matrix approach fundamentally different than the diagonalization approach?

I still have many open questions, and these were just about matrices!

Andrew Robbins

PS. @bo198214:
I still need to do some research on the (b>1) thing, but my current ideas can be summarized by this thread which I intend on updating as soon as I have better ideas...

[Gottfried]

Well, my intention is simply to reflect, to give an overview where "we are", "what we have" after several monthes of intense experimentation and discussion. Is it already worth to communicate this to the wider open? Meanwhile, it came to my mind to focus on an "open workshop" or conference (online or at a university) to present and discuss our findings and open problems, which show up to be the problems at core - something like that.

I'd also appreciate to systematize the faq and a ref in Andrew's style.

With the matrix method

I am still having difficulty with the regular/matrix approaches. I have read Gottfried's approach many times, and each time I understand a little bit more, but I must admit that I find Aldrovandi's discussion of matrices seem to be much more understandable.

- well, as I understood, it is only a transposed version of the widespread diagonalization methods; just basic matrix-algebra. For the integer case powers of matrices as notation for operations on the powerseries, which are needed to describe higher iterations. I always say "my matrix method" because I saw functional-entries in matrices, when discussed in connection with Bell/Carleman-entries and I was/am not 100% sure, whether I got that functional entries right and whether they in fact reflect the same things as in my matrices. But as Henryk pointed out several times, I just implemented regular iteration on powerseries-based functions.

If participants are ready to engage with some time, I could imagine to install a tetration-forum-matrix-workshop to explain the basic matrix-operations in specific questions and answers, although there is nothing special in here.