Consider the analytic continuation of the factorial function $n!$ given by $\Gamma(z)$ (note $n!=\Gamma(n+1)$), and suppose $z=a^{-n}$, where $a,n\in\mathbb{N}$ are positive integers.

Are there any known properties/identities/functional identities of the Gamma function in this case? I have searched the web but cannot find anything at all.

  • $\begingroup$ $a^{-n}$ is not necessarily integer, so what do you mean by $z\equiv{a^{-n}}$? $\endgroup$ – barak manos Jul 9 '14 at 12:48
  • $\begingroup$ @barakmanos sorry for the confusion. I meant that the argument is the reciprocal of an integer base and exponent. $\endgroup$ – Pixel Jul 9 '14 at 12:52
  • 1
    $\begingroup$ A somewhat trivial property for $a>1$: Because $0< a^{-n} < 1$ you have $\Gamma(a^{-n}) > 1$ $\endgroup$ – gammatester Jul 9 '14 at 12:56

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