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I was trying to convert an expression with the gamma function to an expression with the factorial function, which is merely gamma(x+1) and (x-1)! is just gamma(x), but somehow everything I thought I knew about it, after confirming it on a computation engine, is now broken and reality is falling apart for me.

I had $$\Gamma\left(\frac x 2 + 1\right)$$ which I then manipulated into $$\Gamma\left(\frac{x+2} 2\right)$$ and from there I thought I converted it into a factorial by simply reciprocating another factor of the expression on the outside, like so $$\Gamma\left(\frac{x+2}2\right)=\frac 2 {x+2}\left(\frac{x+2} 2\right)!$$ but somehow excel says that's wrong.

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  • $\begingroup$ Yes that seems to be correct (given that $x$ is even of course so that the factorial is an integer). What exactly is excel saying is wrong? Not sure how it handles a factorial of a non-integer. $\endgroup$ Commented Jun 9, 2017 at 3:48
  • $\begingroup$ can you please explain how excel says its wrong? $\endgroup$
    – Kartik
    Commented Jun 9, 2017 at 3:49
  • $\begingroup$ It's just giving me like all kinds of odd decimal numbers on integer inputs, decimals that seem to be increasing in a small but linear fashion. $\endgroup$
    – RayOfHope
    Commented Jun 9, 2017 at 3:54

1 Answer 1

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For any natural number $n$, $$\Gamma(n)=(n-1)!$$ Therefore, if $\frac x 2+1$ is a natural number (i.e. if $x$ is a nonnegative even integer), $$\Gamma\left(\frac x 2+1\right) = \left(\frac x 2 \right)!$$ Which is equivalent to the answer you got.

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  • $\begingroup$ That doesn't really make sense to me. If $$\Gamma(x)*x=\Gamma(x+1)=x!$$ then it should follow that $$\Gamma(x)=\frac{\Gamma(x+1)}{x}=\frac{x!}{x}$$ $\endgroup$
    – RayOfHope
    Commented Jun 9, 2017 at 3:53
  • $\begingroup$ I corrected my error. The answer you got is correct. $\endgroup$
    – florence
    Commented Jun 9, 2017 at 3:54
  • $\begingroup$ Oh okay awesome good to know I'm not clinically insane. $\endgroup$
    – RayOfHope
    Commented Jun 9, 2017 at 3:55
  • $\begingroup$ But do note that everything here only makes sense if $x$ is even. If $x$ is odd, then then $x/2$ is not an integer, and so the usual definition of the factorial function does not apply to $x/2$. This may be where you're getting the error. $\endgroup$
    – florence
    Commented Jun 9, 2017 at 3:57
  • $\begingroup$ The cause might be related to that. I would expect excel to calculate the factorial function as gamma(x+1). However, if what you were saying was completely true, excel would tell me "ERROR," it shouldn't give me any number at all if it didn't know how to calculate factorials at fractional values. $\endgroup$
    – RayOfHope
    Commented Jun 9, 2017 at 3:58

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