Another way to write $ \frac{\Gamma(\frac{1}{2}-x)}{\Gamma(x)} $

Question

How can I evaluate the value of $$ \frac{\Gamma(\frac{1}{2}-x)}{\Gamma(x)} $$

Is there a simple way to write this term, (Simple in the sense that there is no denominator term)?

I have tried to use the duplication and the reflection formula of the Gamma function, but it doesn't help much. Maybe they need to be applied multiple times. Any ideas\suggestions are appreciated.

Answer 1

Hint

Use $$\frac{\Gamma \left(\frac{1}{2}-x\right)}{\Gamma (1-2 x)}=\sqrt{\pi }\frac{ 2^{2 x}}{\Gamma (1-x)}$$ and then the duplication or reflection formula.

Answer 2

For getting a representation involving zeta functions use

$$\frac{\Gamma\left(\frac{\xi}{2} \right)}{\Gamma\left(\frac{1-\xi}{2} \right)}=\frac{\zeta(1-\xi)}{\zeta(\xi)}\pi^{\xi -1/2}$$

for some $\xi \in \mathbb{C}$ with $Re(\xi) >0$