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.

  • 1
    $\begingroup$ it is a nice question but meaningless $\endgroup$ Jul 13, 2015 at 17:26
  • 2
    $\begingroup$ It's very easy to write down functions that satisfy only the first functional equation. You can specify any behaviour you want for $x \in [1,2)$ and then you'll have a well-defined function for $x \geq 1$ by that equation. The other two are more restrictive in ways that are a bit more difficult to describe. I suppose you could specify $f(z)$ for $z > 1/2$ in the last one arbitrarily, and extend it to the rest of the real line through the functional equation as well. $\endgroup$
    – davidlowryduda
    Jul 13, 2015 at 17:28
  • 1
    $\begingroup$ Some alternatives to the gamma function are described here: luschny.de/math/factorial/hadamard/… $\endgroup$ Jul 13, 2015 at 17:48


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