Real-analytic tetration uniqueness criterion?

[tommy1729]

@Tommy1729:

The Gamma function is different from tetration. To make an alternative tetration, use

\( \mathrm{tet}^{*}(z) = \mathrm{tet}(z + \theta(z)) \)

where \( \theta(z) \) is a 1-cyclic function with \( \theta(0) = 0 \).

To make an alternative "Gamma function", use

\( \Gamma^{*}(z) = \Gamma(z) \theta(z) \).

where \( \theta(z) \) is a 1-cyclic function with \( \theta(0) = 1 \).

This is because the functional equation for tetration is

\( \mathrm{tet}(z + 1) = \exp(\mathrm{tet}(z)) \)

whereas that for the Gamma function is

\( \Gamma(z + 1) = z \Gamma(z) \).

Take the ratio of two solutions of this equation, and you will see it is 1-periodic. Thus, a 1-periodic multiplication factor (unlike for tetration, where you need composition!) will convert one "Gamma-like function" into another.

Also,

\( \Gamma((z + 1) + \theta(z + 1)) = \Gamma(z + \theta(z) + 1) = (z + \theta(z)) \Gamma(z + \theta(z)) \ne z \Gamma(z + \theta(z)) \).

Yes of course you are right.
Sorry silly of me.

But the argument still remains ...

What is your opinion about the periodic functions that I tried ?
Did you try those ?

regards

tommy1729