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Transseries, nest-series, and other exotic series representations for tetration
(11/26/2009, 03:57 PM)Daniel Wrote: Howdy,
Thanks for the information on transseries, I hadn't heard of them before. The most general method I've developed for extending tetration is based on using a system of nested summations like you are talking about. See
and .

Yes. The problem with that method seems to be that it appears to be the so-called "regular iteration" of a function near a fixed point. This works good for tetration to bases but not outside that range, where it generates an entire tetrational, which is not good because such a thing is not real at the real axis for, e.g. , and inconsistent with the behavior of tetrational in that range, where it has singularities. This seems to be a pitfall of every fixed-point iteration method I've seen so far, unless I've missed something...

The method I mentioned, however, does not approach the problem from the direction of dynamical systems, but rather from that of superfunctions, functional equations, algebra and analysis, though this may overlap, and does not involve fixed points.

It uses what I call "Ansus' formula" for the tetration (after the poster who first posted it here):

The way this is derived is from the recurrent equation of the iteration




By differentiating the recurrent equation we get a recurrent equation for the derivative, which can be turned into an "indefinite product", which can then be turned via an exp/log identity (see here: into a sum, and the result is the above formula with a sum from 0 to . The idea is then to generalize this to fractional, real, or complex values of , obtaining what I call "continuum sum", but it could also be called "fractional sum", "sum with non-integer bounds", and "continuous antidifference".

Now obviously there are many ways to interpolate a sum, just as there are to interpolate tetration directly, however, the Ansus formula transfers the problem of tetration into the domain of sums, which are an easier operation to study. So if we have an interpolation for sums, we should be able to use that formula to get one for tetration.

The trick is, what is the most "natural" interpolation of a sum? For power functions, we have a formula called "Faulhaber's Formula". It turns out one can actually get to this formula after starting with just basic identities for sums (just ask if you want to see the method), plus some analytic continuation. Thus the possibility of applying that to an analytic function given by a Taylor series presents itself, as it's a sum of powers.

The problem, however, is that as mentioned it only appears to work for a restricted subset of analytic functions, which seem to imply (though I do not yet have the rigorous proof, but basically the coefficients must decay more rapidly than the Bernoulli numbers grow) the function must be entire and of at most exponential type. Tetration violates both requirements.

However, perhaps Faulhaber's formula is not as useless as it may seem here. There are much more general series representations than just Taylor series, including the transseries I mentioned, which is actually quite general in itself: it includes Taylor series, and also other types of expansions such as nested power series, "exp-series" (like what I gave for the double-exponential function), Newton series, Mittag-Leffler star expansion, and more. They can have larger regions of convergence than Taylor series, such as a whole half-plane and even an entire star like the Mittag-Leffler star (e.g., the Mittag-Leffler star expansion (see here:, but it doesn't provide the necessary magic coefficients and I've had an awful time trying to find them.). As could be seen from the first post, the use of a properly-chosen transseries can provide the continuum sum of even more functions than with the direct application of Faulhaber to Taylor series. (two examples were given in my opening post, that of the double exponential function and that of the reciprocal function, neither of which Faulhaber-on-Taylor could sum due to the first being not of exponential type and the second having a singularity (so not entire))

For extending tetration, the problem is then finding a good transseries representation and defining continuum sum, exp, integral, and mult by operators so they'll converge for the tetration series, and also such that the iteration of Ansus' formula converges. And so far, I don't have much on this.

Messages In This Thread
RE: Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/27/2009, 01:29 AM

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