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Transseries, nest-series, and other exotic series representations for tetration
That's right. The trick here is finding a representation of a given analytical function as some series of terms that can be continuum-summed (either they should be continuum-summable via direct application of Faulhaber's formula, or they should be reducable to some alternative representation that is summable via Faulhaber's formula (or some chain of such reductions)) so that the continuum sum series also converges, and even better if that continuum sum and the integral can be represented in the same form (so as to enable an iterative map on the series from the continuum-sum tetration formula). That's why I mentioned the "transseries", which gives one possible approach. Transseries actually forms a somewhat complicated theory and there's a lot of stuff in that paper, but it includes more sophisticated and exotic forms of series than just simple power series, such as nested sums of power series, exp-series, polynomial-series (incl. Newton and Mittag-Leffler series in a star), and more.

So far, I've found 2 types of transseries representation such that if the series converges, the continuum sum does as well (proof given earlier here):


Newton series or umbral series(*):


where is the falling factorial. If can be given by a Newton series on some part of the plane, then

(where is the forward difference operator.)

(*) I call this umbral series because this series is akin to a Taylor series with finite differences, see e.g. "umbral calculus".

However neither are very useful for the extension of tetration. The exp-series always have a periodicity of , thus they cannot represent aperiodic functions or functions of different period. Also, the exp-series does not continuum-sum to another exp-series, as one should be able to see from the above. The Newton series can only represent functions with that grow at most exponential-type -- tetration grows too fast. Tetration at bases does, however expand to a Newton series, and we don't even need to run the sum formula to find this out. I think this is equivalent to the regular iteration. The sums for and can be recovered from Faulhaber's formula.

Another possible series is the Mittag-Leffler star expansion, from here:

(special case for expansion at 0)

(appl. Faulhaber's formula, so are the Bernoulli polynomials)

where , which are the Taylor coefficients of . This series converges in the entire Mittag-Leffler star of the function. Whether the continuum sum converges will likely depend on the characteristics of the magic numbers , which I have not been able to find. The site mentions they can be "evaluated once and for all". Two refs were given. I was emailed a snapshot of the proof from the book by Markushevich, but it did not detail how to actually obtain the coefficients. The second reference by Borel was in French, which I don't know, so that was not very useful.

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