quickfur wrote:Now, this is an area that I'm deeply interested in.

Hehey, we are beating the 3, regardless whether as number of dimensions or as number of operations!

.For a long time, I've been interested in generalizing the exponentiation function to tetration

Ok, here I should add some notes.

1. Recently Galidakis showed that there is a continuous (even arbitrary times differentiable) extension of the higher operations defined by the original Ackermann function. The problem with the extension is that it is quite arbitrary, a uniqueness criterion is lacking.

2. I dont see a reasonable connection, regarding a continuous extension, between the extension of x^(x^...) and the extension of e^(e^....x). The first one has nearly no clues how to define for example half a tetration, while the latter has a clue, i.e. f(f(x)) = e<sup>x</sup>.

3. If we would of course define f(x):=x<sup>x</sup> then we could try to find the analytic iterates (as introduced in my previous post), because f is analytic and has a fixed point at 1. Look at:

f(f(x)) = (x<sup>x</sup>)<sup>x<sup>x</sup></sup> = x<sup>x<sup>x</sup>x</sup>

f(f(f(x))) = x<sup>x<sup>x<sup>x</sup>x</sup></sup><sup>x<sup>x</sup>x</sup>

We see that the tower is n+1 high, for n iterations of f. Iteration of f can also be done by iteration of g(x):=xe<sup>x</sup> (which is the inversion of the famous Lambert W function), if we notice that g = ln o f o exp and thatswhy g<sup>n</sup> = ln o f<sup>n</sup> o exp. g has even the desired g(0)=0 standard fixed point. The analytic iterate should be unique, though I didnt elabortate yet whether the previously given analytic iteration formula converges for g at 0. If not then at least for each iterate is a unique analytic function defined on x>0 that approximates the analytic iteration power series in 0.

The iteration of f was indeed regarded already by Frappier in "Iteration of a kind of exponentials", 1990, though it seems he didnt show neither uniqueness nor analyticity of the iterates. So this work shall be done by the one or another.

4. e<sup>x</sup> has a family of real analytic iterates (as showed by Kneser already 1949), unfortunately these are not unique either, because the development is around a complex fixed point. The quite reputable mathematician Szekeres devoted much time to finding a "best" iterate, but I would say still unsuccessfully (see his "Abel's equations and regular growth ....", 1998 (and notice that he was born 1912!)).

Why havent we such uniqueness problems for the lower operations multiplication, and exponentiation?

The short answer is because (nm)x = n(mx) and x<sup>nm</sup> = (x<sup>n</sup>)<sup>m</sup>. Because by this law we have a clue how to define (1/n)x or x<sup>1/n</sup>. This law should then be also valid on the extension the rational numbers: x=x<sup>(1/n)n</sup> = (x<sup>1/n</sup>)<sup>n</sup> and hence it is clear that x<sup>1/n</sup> must be the inversion function of x<sup>n</sup>. Similarly for the multiplication, though not so interesting.

Unfortunately this law is no more valid for tetration, i.e. usually x^^(nm) != (x^^n)^^m, because the parenthesizing is different (which matters for x<sup>y</sup> in contrary to multiplication and addition, which are associative). To come out of this dilemma I invented a new number domain "arborescent numbers" in which this law is valid for all higher operations. And I am frolicking that one time all the higher operations have a unique continous extension there. The current state of my research can be found here.