According to wikipedia: https://en.wikipedia.org/wiki/Functional_equation
The $\Gamma$ - function is the only solution that suffices the following three equations:
$$ f(x)={f(x+1) \over x}\tag{$*$} $$
$$ f(y)f\left(y+\frac{1}{2}\right)=\frac{\sqrt{\pi}}{2^{2y-1}}f(2y) \tag{$**$}$$
$$ f(z)f(1-z)={\pi \over \sin(\pi z)} \tag{$***$} $$
Now I do wonder, how functions look like that suffices just 1 or exactly 2 of the given equations above, if there are really examples of such functions, that are more intuitive. How would one approach that problem?
Every constructive comment or post is appreciated.