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Bold and Disappointing Experiment With Hermite Polynomials!!!

Long time no see.

I have recently come back to try another whack at tetration after such a long time away from the subject, this time going for the bold gold -- an explicit, analytic formula for the tetrational to base , that is, the Kneser tetrational.

I dusted off the Hermite polynomial "gadget" that was mentioned here:

with a long and wild and crazy idea to ... drumroll ... attempt to actually solve the continuum sum analytically to obtain an explicit series formula for the tetrational function. Unfortunately, the results from this latest endeavor look to be disappointing, but they also make one wonder about the nature of the continuum sum formula itself, and in particular, possible (non-)uniqueness considerations regarding the solution.

To recap. First off, a long time ago, for those who have not heard it, a poster that used to be here called "Ansus" suggested that tetration could be expressed using the following weird formula:


This is called "weird" because the right hand side has a sum whose upper bound may not be an integer. The key to using the formula, then, is to suitably generalize the summation operator so as to admit a non-integer upper bound, in a suitably "natural" way. The idea here being that it is easier to generalize summation than to generalize iterated functions, as summation is as a neater, more well-behaved operation.

Now, this author fell in love with this formula at the drop of a hat. This author had experimented with the idea of this kind of "generalization of sum", which he now calls "Continuum sum", before, and got results, and so this could be a very interesting method.

As a way to illustrate the possibilities, consider the well-known formula


Clearly, there is nothing stopping one from plugging a non-integer into the right. This is a simple generalization. For example, if we take , we get . One can even plug complex numbers, e.g. gives a "sum" of .

One can also do similar to generalize sums of powers , , and so forth -- one them obtains a classic result known as Faulhaber's formula. One might then be led to try and use this to find continuum sums of highly general functions by applying it to a power series expansion, like


The trouble is, however, that if one does this, for many analytic functions the resulting continuum sums do not converge. In particular, if the analytic function has any singularities on the complex plane, or it is entire but with suitably fast growth (much slower than tetration), there will be no convergence.

This led the author to play around with various "divergent summation" methods, also not to much avail since many of them do not permit nice, combinatorial formulae for the series, at best allowing only tantalizing numerical results.

But recently, this method of Hermite polynomials was discovered. Instead of representing a function as a power series, it can be represented as a Hermite polynomial series:


if it is suitably bounded on the real axis. Now tetration is unbounded and singular there, so that won't work. But if we take as hypothesis that the "right" tetrational should be the Kneser tetrational, then it stands to reason it is limited along the imaginary axis, and the formula we would want to use would be


Taking the results from the given link -- which I will not reproduce here for brevity -- we can define a continuum sum for this as

where are Bernoulli numbers.

So now we return to the continuum sum tetration formula:

The exponential is a big problem -- that's a nasty operation on a power series that makes the coefficients into Bell polynomials of the original coefficients, and I have no idea how to work it out for a Hermite series. So to avoid this difficulty we take logs:



Then differentiate once to clear the log:

Now we have to work out what each side is in terms of unknown coefficients for which


First, the continuum sum:


Now we differentiate once. To do this is relatively simple -- the formula for the derivative of a Hermite polynomial is , so by the chain rule the suitable derivative for the one with is and we thus get

We now form the derivative of itself:


For what it's worth, we might as well go a step ahead and take the second derivative too while we're at it, crackin' away:


Now we need to do the Hermite series multiplication. This gets rather damned nasty and UGLY, so to make this easy let us first define


so that



Now we multiply those two azz-whuppin series together, to get

To simplify this further, however, we find we need a formula for the product of two Hermite polynomials. This led to some searching, which led to finding this paper:

Down on page 17, the formula is given, which we translate here as


Thus, we plug that bad boy into the above equation to get

EEGZ! That was a lot of math. Good thing we've got TeX and cut/paste to write it with!

We now set the second-derivative expression equal to this:

Then by manipulation, we get

We now equate coefficients to get

and plugging in the expressions for [math]C_k[/math] and [math]D_n[/math] gives

... and there, our hope dies, pitifully.

This equation system is a non-linear infinite series equation (note the quadratic order terms ) for the respective coefficients ... which means that a solution, if any, is likely to be highly non-unique, and there is no easy way to make a closed form or even a recurrence relation for the . Now it may be there is a unique convergent solution, but I would have no idea how to determine if that is the case (theoretically, there should be least one, the Kneser tetrational, but is there more than one?). It might also be possible to impose some regularity condition on the but again, how we would determine if that would even work at all ... I have no idea.

One possible thought: I am wondering if somehow, the use of the derivative somehow "loses information" as to the solution, which would be another constant only for regular differential equations, but with this continuum-sum thing in there...

Does anyone have any thoughts about all this mess that we just cooked up here?

(Although maybe I made an algebra mistake, which, given all that stuff up there is possible alright, but I don't think it would change the outcome that drastically as to make this solvable. You have the product for sure, and that screws it all up.)

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Bold and Disappointing Experiment With Hermite Polynomials!!! - by mike3 - 01/18/2017, 09:39 AM

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