Gamma function manipulation $\Gamma\left(x+\frac{1}{2}\right) = \text{something}\Gamma(x)$?

Question

Is it possible to write $$\Gamma\left(x+\frac{1}{2}\right)$$ in terms of $$\Gamma(x)?$$ I am currently doing an induction argument and I require this, but haven't been able to figure out a nice manipulation and there is nothing useful on Wikipedia.

Answer 1

If you have a look here, you will find that, for non-negative integer values of $n$, using the reflection formula, we have:$$\Gamma\left(n+\tfrac12\right) = {(2n)! \over 4^n \,n!}\, \sqrt{\pi} = \frac{(2n-1)!!}{2^n} \sqrt{\pi} = {n-\frac12 \choose n}\, n!\,\sqrt{\pi}= \sqrt{\pi}\,{n-\frac12 \choose n} n \,\Gamma(n)$$