The $\Gamma$-function restricted to $(0,+\infty)$ has the following properties:
- $\Gamma(n)=(n-1)!$ for $n=1,2,3,...$.
- The $k$'th derivative $\Gamma^{(k)}$ has no zeros on $(0,+\infty)$ when $k$ is even (starting from $k=0$), and it has exactly one zero when $k$ is odd.
I wonder whether there exists another real analytic (or at least $C^\infty$) function $(0,+\infty)\to\mathbb R$ enjoying these properties. If there are many, I'd like a description of the union of their plots in $(0,+\infty)\times\mathbb R$.