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The $\Gamma$-function restricted to $(0,+\infty)$ has the following properties:

  1. $\Gamma(n)=(n-1)!$ for $n=1,2,3,...$.
  2. 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 $(0,+\infty)\times\mathbb R$.

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    $\begingroup$ Is it possible given the uniqueness of the Gamma function? (see mathoverflow.net/questions/23229/…) $\endgroup$ Aug 2, 2022 at 17:53
  • $\begingroup$ Not if the properties 1 and 2 imply the hypothesis of the uniqueness theorem, i. e., log-convexity and the functional equation. But how can we prove that? $\endgroup$
    – igorf
    Aug 2, 2022 at 18:58

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