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