Gamma function and tetration

[sheldonison]

....I also notice that the Gamma function can be uniquely specified by Wielandt's theorem, which uses the condition of being bounded in a strip parallel to the imag axis along with holomorphism on the right halfplane and the starting value).

Thanks for sharing your observations about the Gamma function, and Wielandt's theorem. I looked up Wielandt's theorem, which is very nice, although it is probably not directly applicable to tetration. I also like your observation that for tetration from the alternative fixed point, the inverse function has branch points. So if we require sexp(0)=1, and sexp'(0)<>0, then sexp(z) is for the primary fixed point(s). This also rules out the alternative fixed point solution, (for now).

...But I wonder whether the strip-bounding condition (plus holomorphism at at least the right half-plane and also that \( \mathrm{tet}(0) = 1 \)) at least determines the solution up to a possible choice of fixed points. Might it? As then the rest of the uniqueness would be simple: just add that the function should approach the two principal fixed points at \( \pm i\infty \), and then we'd have a full uniqueness specification for the tetrational (and this could probably also be generalized to functions defined via other recurrences as well).

We additionally require that sexp(z) be analytic in the upper/lower halves of the complex plane, with singularities only at the real axis.

Let us say we have sexp(z) defined in a strip, approaching the two principal fixed points at \( \pm i\infty \). Then, if we have another purported solution, f(z) that also has \( f(z+1)=\exp(f(z)) \), also with f(0)=1, and f'(0)<>0, then f(z) can also be defined in the same strip. Take \( \theta(z)=\text{sexp}^{-1}(f(z))-z \), which is defined not only in that strip, but everywhere in the complex plane. If theta(z)=0, then f(z)=sexp(z) and we are done. if theta(z)<>0, then theta((z) may have singularities, in which case if f(z) also has singularities* where as sexp(z) is analytic in the upper and lower halves of the complex plane, then we are done. If theta(z) does not have singularities, then it can be shown that somewhere in the strip, theta(z) gets arbitrarily large negative imaginary due to exponential growth, \( - i\infty \) so that \( z+\theta(z) \) also gets arbitrarily large negative imaginary, so f(z) at \( + i\infty \) must not approach only the fixed point, since f(z) will also get arbitrarily close to the other fixed point at \( + i\infty \), which is a less than ideal definition for tetration.
- Sheldon
* the case with \( \theta(z) \) singularities is complicated, so this is not a proof, because \( \text{sexp}(z+\theta(z)) \) must be shown to not cancel out the theta(z) singularity.