how to find $\Gamma(n+3/2)$ I'm newly introduced to the gamma function. I was wondering how can I calculate: $$\left(n + \frac 12\right)!$$
When I entered the above in wolfram alpha the result was: $$\Gamma\left(n + \frac 32\right)$$
After researching about the gamma function, I tried to prove that the two above expression are the same but I failed. So how can I calculate $\Gamma\left(n + \frac 32\right)$?
 A: Wikipedia gives
$$
\Gamma \left({\tfrac {1}{2}}+n\right)={\frac {(2n-1)!!}{2^{n}}}\,{\sqrt {\pi }}={\frac {(2n)!}{4^{n}n!}}{\sqrt {\pi }}
$$
To get $\Gamma \left(n+{\tfrac {3}{2}}\right)$, use $n+1$ instead of $n$ in this formula.
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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\begin{align}
\Gamma\pars{n + {3 \over 2}} & =
\Gamma\pars{\bracks{n + 1} + {1 \over 2}} =
{\Gamma\pars{2n + 2} \over
\pars{2\pi}^{-1/2}\, 2^{2n + 3/2}\,\Gamma\pars{n + 1}}
\end{align}
In the last step, I used the
Gamma $\ds{\Gamma}$ Duplication Formula.
Then,
\begin{align}
\Gamma\pars{n + {3 \over 2}} & =
\bbx{\large{\root{\pi} \over 2^{2n + 1}}\,{\pars{2n + 1}! \over n!}} \\ &
\end{align}
