# What did I calculate wrong with the gamma -> factorial?

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.

• 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. Commented Jun 9, 2017 at 3:48
• can you please explain how excel says its wrong? Commented Jun 9, 2017 at 3:49
• 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. Commented Jun 9, 2017 at 3:54

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.
• That doesn't really make sense to me. If $$\Gamma(x)*x=\Gamma(x+1)=x!$$ then it should follow that $$\Gamma(x)=\frac{\Gamma(x+1)}{x}=\frac{x!}{x}$$ Commented Jun 9, 2017 at 3:53
• 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. Commented Jun 9, 2017 at 3:57