Problem related to Gamma ($\Gamma$) function.

Question

I have to prove the following equation

$$\frac{\Gamma(\frac{m}{2})\Gamma(\frac{m+1}{2})}{\Gamma(\frac{n+m}{2})}=2^n\times\frac{\Gamma(m )\Gamma(\frac{m+n+1}{2})}{\Gamma(n+m)}$$

I started by considering the RHS of the above equation.

I'm not getting how to

• put denominator 2 under $(n+m)$ of RHS
• get rid of $\frac{n}{2}$ from the $N^r$ of RHS.

Is there any formula(Linearity type) which can help me adjust powers of 2 to get LHS?

Answer 1

Using the duplication formula for $\Gamma(m)$ and $\Gamma(n+m)$, we have $$\frac{\Gamma(m)\Gamma(\frac{m+n+1}{2})}{\Gamma(n+m)}=\frac{\color{DarkBlue}{2^{m-1}\Gamma(\frac{m}{2})\Gamma(\frac{m+1}{2})}\color{gray}{[/\sqrt{\pi}]}\Gamma(\frac{m+n+1}{2})}{\color{DarkBlue}{2^{m+n-1}\Gamma(\frac{n+m}{2})\Gamma(\frac{n+m+1}{2})}\color{gray}{[/\sqrt{\pi}]}}=\frac{\Gamma(\frac{m}{2})\Gamma(\frac{m+1}{2})}{2^n\Gamma(\frac{n+m}{2})}.$$