Hi.

I noticed that the "gamma function" , the continuous version of the factorial, i.e. continuum solution of

in the complex -plane, obeys the following law:

similar to what Tetration does:

.

Is there something interesting here?

(and ditto for the lim to )

Indeed, if you plot the two on graphs, they're kind-of similar to each other.

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). Now it seems this does not work entirely well for tetration, since here:

http://math.eretrandre.org/tetrationforu...452&page=2

an alternative strip-bounded solution, using different fixed points (but which is not as well-behaved, namely the inverse function has branch points at the real axis), was constructed. But I wonder whether the strip-bounding condition (plus holomorphism at at least the right half-plane and also that ) 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 , 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).

I noticed that the "gamma function" , the continuous version of the factorial, i.e. continuum solution of

in the complex -plane, obeys the following law:

similar to what Tetration does:

.

Is there something interesting here?

(and ditto for the lim to )

Indeed, if you plot the two on graphs, they're kind-of similar to each other.

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). Now it seems this does not work entirely well for tetration, since here:

http://math.eretrandre.org/tetrationforu...452&page=2

an alternative strip-bounded solution, using different fixed points (but which is not as well-behaved, namely the inverse function has branch points at the real axis), was constructed. But I wonder whether the strip-bounding condition (plus holomorphism at at least the right half-plane and also that ) 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 , 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).