How Do You Solve Gamma Function Identities?

[binbagsss]

Homework Statement

To show:

Homework Equations

The Attempt at a Solution

To be honest, I'm pretty stuck.

I could try to use the third identity:
$\Gamma(-k+\frac{1}{2})=\frac{2\sqrt{\pi}}{2^{-2k}}\frac{\Gamma(-2k)}{\Gamma(-k)}$

but this doesn't really seem to get me anywhere.

I could also try to use the first identity, by adding and subtracting a 1/2:

$\Gamma(1+(-k-\frac{1}{2}))=(-k-\frac{1}{2})\Gamma(-k-\frac{1}{2})$

Which again doesn't seem to help..

Thanks in advance.

 

[fresh_42]

Homework Statement

To show:

View attachment 211112

Homework Equations

View attachment 211111

The Attempt at a Solution

To be honest, I'm pretty stuck.

I could try to use the third identity:
$\Gamma(-k+\frac{1}{2})=\frac{2\sqrt{\pi}}{2^{-2k}}\frac{\Gamma(-2k)}{\Gamma(-k)}$

but this doesn't really seem to get me anywhere.

I could also try to use the first identity, by adding and subtracting a 1/2:

$\Gamma(1+(-k-\frac{1}{2}))=(-k-\frac{1}{2})\Gamma(-k-\frac{1}{2})$

Which again doesn't seem to help..

It (the last version) helps a lot if you combine it with an induction argument.