Uniqueness Criterion for Tetration

[jaydfox]

They suffice neither for the Gamma function nor for exponentiation nor for multiplication. The criterion that makes the Gamma function unique is logarithmic convexity and the criterion that makes exponentiation and multiplication unique is the translation equation (see the FAQ for a description of both).

So - as you said - for uniqueness there shall be found a suitable criterion, (however I beleave there exists nothing suitable XD).
For the fractional iteration of functions there are uniqueness criterions at hand if the function has a fixed point. Unfortunately this is not the case for exp, indeed there is no uniqueness criterion for continuous iteration of exp too. (For the relationship between iteration of exp and tetration see also the FAQ.)

Well, I'm not sure if we can define a uniqueness criterion, but we can impose certain limits. For example, it seems very desirable to me that fractional exponential functions should be convex. For example, if we define F(F(x)) = e^x, then it seems sensible to me that F(x) should be convex. Ditto for G(G(G(x))) = e^x, and so on.

Well, as I'm sure we're all aware, for qth iterate of e^x, we can define the function as parameterized curve:

\( R(t)\ =\ < ^{t} e,\ ^{t+q} e >
\\ \ \text{or}
\\
R(z)\ =\ tet_e(tet_e^{\tiny -1}(z)+q) \)

Pushing forward to real iterates (rather than just the "well-defined" rational iterates), let's use r instead of q. Differentiation then tells us that:

\( \begin{eqnarray}R^\prime(z) & = & tet_e^\prime(tet_e^{\tiny -1}(z)+r)\ \times\ \left(D_z\ tet_e^{\tiny -1}(z) \right) \\
& = & tet_e^\prime(tet_e^{\tiny -1}(z)+r)\ \times\ \left(\frac {1}{tet_e^\prime(tet_e^{\tiny -1}(z))} \right)\\
& = & \frac {tet_e^\prime(tet_e^{\tiny -1}(z)+r)}{tet_e^\prime(tet_e^{\tiny -1}(z))}\\
& = & \frac {tet_e^\prime(Z+r)}{tet_e^\prime(Z)}\end{eqnarray} \)

My calculus is a bit rusty, so someone spot-check the conversion from:
\( D_z\ tet_e^{\tiny -1}(z) \\
\ \text{to} \\
\frac {1}{D_z\ tet_e(tet_e^{\tiny -1}(z)) \)

(I'm pretty sure that's an identity, but please let me know if I'm wrong.)

From here, we need to guarantee that R'(z) is always increasing. To do this, it suffices to show that ln(D_z tet_e(z)) is convex (I don't have time to post the explanation, I'll get back to that after work). Which in turn simply means that:

\( D_z^2\ ln(tet_e^\prime(z))\ \ge \ 0 \)

This in turn is equivalent to saying that the first derivative of tet_e(z) is log-convex. It turns out that my formula meets this criterion, so it's not a uniqueness constraint, but it can help sift out the really "wrong" solutions. Additionally, while the first derivative of my function is log-convex, a look at the second derivative shows that it's "wavy". (See attached image, which is for my formula that you quoted below.)

   

(Question to forum experts: Is there a way to attach a full-sized image, or perhaps enlarge the thumbnail that displays for attachments?)

Can you explain your formula
\( T(x,\ y,\ n) = \left{
\begin{eqnarray} \alpha_0\ +\ y\ +\ x^{T(x,\ y-1,\ n-1)} & , & n\ >\ 0 \\ \alpha_0\ +\ y & , & n\ =\ 0 \end{eqnarray} \right.
\\ \
\\ \
\\ \
\\
{\Large ^y x}\ =\ \lim_{m,n\to\infty}{ln^{\small (m)}T(x,m+y,m+n)} \)
in a bit more detail?

PS: For not using TeX before your post was amazing

I'll get back to this question after work tonight.

Edit by bo198214: I inlined the image.