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"

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"

Gottfried Helms, Kassel