Is there a Smooth, convex alternative to the Gamma fucntion.

Question

Is the a function $f(x)$ other then the Gamma function with said properties.

1. $f(x)=(x-1)!$ when x is a non-negative integer.
2. $f(x)$ is smooth (infinitely differentiable.)
3. $f(x)$ is convex.
4. $f(x)=xf(x-1)$. for x>1

I know that the Gamma function is the only solution if 3. is strengthened to being Logarithmically convex.

Answer 1

You might try Hadamard's gamma function? $$ H(x) = \frac{1}{\Gamma (1-x)}\,\dfrac{d}{dx} \left \{ \ln \left ( \frac{\Gamma ( \frac{1}{2}-\frac{x}{2})}{\Gamma (1-\frac{x}{2})}\right ) \right \} $$ This looks reasonably smooth and convex (not sure about the bump and have no proof of this). But.. it only satisfies your functional relationship for positive integers $x$ and is otherwise $$ H(x+1) = xH(x) + \frac{1}{\Gamma(1-x)} $$ all credit to Wikipedia.