Does anyone think that tetration by a non-integer will ever be defined ... really properly?

Great mathematicians struggled with finding an implementation of the non-integer factorial for a long time to no avail ... and then eventually Leonhardt Euler devised a means of doing it, & by a sleight-of-mind that was just so slick & so simple in its essence ... and yet so radical! Is there any scope for another such sleight-of-mind as that, whereby someone might do similarly for tetration, or have they all been used-up? It's impossible to conceive of how there might be any truly new conceptual resource left of that kind. But it's actually tautological that that is so, because the kind I am talking about is precisely the kind that is essentially radically new, & of even how it might be thought hitherto unconceived!

But the attempts I have seen so far at defining tetration by general real number look to me for all the world like mere interpolation - some of them indeed very thorough & cunning & ingenious (insofar as I can follow them atall) - but lacking that spark of essential innovation that is evinced in Euler's definition of the gamma function.

Just incase it seems I have gotten lost in philosophy, I'll repeat the question: will there ever be a definition of tetration by general real number that resolves the matter as thoroughly as Euler's definition of the gamma-function resolved the matter of factorial of general real number?

  • I am no expert on history, but googling yields the name "Leonhard Euler." I am unsure where you get the "t"... – Mohammad Zuhair Khan Nov 21 at 16:25
  • I once read the appendix to TE Lawrence's Seven Pillars of Wisdom ... I think it was a bad influence on me! – AmbretteOrrisey Nov 21 at 16:35

We're trying, but it's hard.

A better analogy than the Gamma function would be the way you can now define $x^y$ for real $y$. Why does $x^{3/2}$ make sense? Because I can solve $y^2=x^3$. (Admittedly any solution $y=y_0$ implies $y<y_0$ is a solution too, but we have a convention to get around that for $x>0$.)

So what would $^{3/2}x$ mean? Presumably, a solution of $y^y=x^{x^x}$. Unfortunately, the values of $y^y$ for $y\in (0,\,\frac{1}{e})$ are repeated again for some $y>\frac{1}{e}$ in a... not particularly simple way, so it's already getting confusing. There are similar headaches when trying to define $^{k/(2l)}x$ for $x>0,\,k,\,l\in\mathbb{N},\,2\nmid k$.

I'm not sure whether you can even prove $^y x$ with $y$ irrational can be defined by continuity, i.e. whether we can prove any rational sequence $y_n$ with $\lim_{n\to\infty}y_n=y$ gives the same $n\to\infty$ limit of $^{y_n}x$.

Having said all that, I bet we'll have made a lot of progress within 200 years (even if only in proving what we can't do).

  • I can't answer this as thoroughly as I would like to at the present moment - but one little item stands out - to me, at least, the most fundamental definition of $x^a$ for general real $a$ is that it is the solution of the differential equation $dy/dx =ay/x$ & $1^a = 1$ forall $a$. But I am very fond of that scheme of mathematics inwhich functions are essentially defined by differential equations - as being solutions of them - that as the axiom of what they are. – AmbretteOrrisey Nov 21 at 16:43

The idea of E.Schroeder in the 19'th century for bases $b$ (allowing two real fixpoints for iterations, for instance $b=\sqrt{2}$) , sometimes called "regular iteration" seems to me a real good one. It allows a meaningful expression for fractional iteration from some starting point $z_0$ towards some endpoint $z_h$ where $h$ means the (possibly fractional, even complex) iteration-(h)eight, such that $z_0$ is the initial value, $z_1 = b^{z_0}$ is the first (integer) iteration and so on.

However, that method needs conjugacy to be able to be applied to all real $z_0$ : one has to choose the appropriate fixpoint for shifting the power-series for the exponentiation with base $b$ and get an evaluatable analytical answer at all.
Unfortunately it has been observed, that in the cases $z_0$ where each fixpoint-conjugacy can be taken, the results of fractional iteration are different - even if only by some 1e-25 or so.

A basic problem for the general $b$ is the multivaluedness of the complex exponentiation/logarithmization, or say, the "clock-arithmetic" with respect to the $2 \pi î$-term in the exponentiation - I have once seen an article (R.Corless & al.) on the "winding number" which tries to make sense of introduction of one more parameter for the complex numbers to overcome that "clock-arithmetic" to a real-arithmetic. But that was no real progress for the problem here.

So I think, similarly to the full workout of L. Euler about the multivaluedness of the logarithm and then the representation of the gamma-function as an infinite sum of partial products, we need some more idea here - the E. Schroeder-idea seems to me just like a small insular solution, however nice ever...

(just as a remark: you might consider the two common versions of the interpolation of the Fibonacci numbers $fib(n)$ to continuous functions of the index: there is one ansatz providing real numbers for real index, and another ansatz (in analogy perhaps to the Schroeder method) which gives complex numbers for real indexes but seems more smooth when seen overall in the complex plane. See a small essay about this at my homepage)

  • I presume that by "the first ansatz" you mean the φⁿ±φ⁻ⁿ thing. I have nod idea about the other, but I am very curious about it now, especially as you say it is an instance of the Schroeder method ... which sounds like it could do with 'tapering' by means of a nice familiar & relatively simple example for someone coaching themself in it from the beginning! Anyway - thank-you for the information ... but it dashed my hopes a bit when you began to speak of it being yet another insular effort. But I think (i) it does require fundamentally new thought (ii) it is there - yet to be found! – AmbretteOrrisey Dec 7 at 20:36
  • @AmbretteOrrisey: thanks for your comment. Unfortunately I'm much busy this and next days and have no space to step in again. Let's see end of next week... – Gottfried Helms Dec 8 at 9:12
  • Of course I don't expect you to dispense me a course on the Schroeder method! On the contrary, thankyou for taking the trouble to steer my attention in that direction. And I do often just throw ideas out without the expectation that those who catch them process or develop them for me! – AmbretteOrrisey Dec 8 at 12:31
  • @AmbretteOrrisey: I've found my old discussion about the interpolation of the fibonacci numbers, I've added the link to my remark in my answer. First I thought I'd posted that as Q&A here in MSE but I did not know this place here and communicated via the sci.math-newsgroup in the usenet. Hope the essay is helpful/explanative about my remark. – Gottfried Helms Dec 9 at 2:47
  • Looks ike you're giving me a course of instruction anyway! That's your work? I think I can discern your style of writing in it. I've only just looked at it - it'll take a while for that to mature. I see the 'other' form of the 'continuous-isation' of the fibonacci numbers - a spiral in the complex plane. Thanks for your ongoing attention ... please don't be tempted to make inroads into your time ... it wasn't meant as a subtle goad or anything when I said I don't expect a course of instruction, and plenty to be fornow.¶ Thanks for that direction - I shall assuredly enjoy delving into that. – AmbretteOrrisey Dec 9 at 8:42

There is a remarkable expression I've found in this connection that might have some bearing on the matter for the iterates the Taylor series of $$\operatorname{f}(x)\to x^{\operatorname{f}(x)}$$ when $x\equiv e^z$, such that $$\operatorname{f_0}(x)\equiv1 ,$$$$\operatorname{f_1}(x)\equiv \exp(z) ,$$$$\operatorname{f_2}(x)\equiv \exp(z\exp(z)) ,$$ etc. The coefficients for $k=0\dots n$ are those of the Lambert W-function; but thereafter, for $k>n$ the coefficients are given by the following recursion. Let $a_{n,0}=1\forall n$, $a_{0,k}=0$ for $k>0$, & thereafter $$a_{n,k}={1\over n}\sum_{j=1}^k ja_{n,k-j}a_{n-1,j-1} .$$ These coefficients are in a sense 'wasted' upthrough $k=n$, inthat they do not actually appear in the Taylor series; and yet they still serve the function of being necessary for the generation of the coefficients that do appear. It is fascinating to my mind, the way there is a kind of discontinuity in the series - the coeffiecients generated by this recursion 'peeling-away' one-at-a -time as $n$ is incremented, 'revealing' the coefficients of the Lambert W-function 'underneath'; and it is a well known result that the limit as the $$\operatorname{f}(x)\to x^{\operatorname{f}(x)}$$ tends to $\infty$ is indeed the Lambert W-function.

Whether this is susceptible of treatment by Schroeder's method I would not venture definitely to say at the present time, as, though I see in outline how that method can be applied to something like the recursion that gives the Fibonacci numbers, I am rather daunted by that discontinuity in the generation of the coefficients in this case; and I cannot see at a glance how it would be encoded.

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