$$\frac{1}{\sqrt{2\pi}}\sigma^m \Gamma\left(\frac{m+1}{2}\right) 2^{\frac{m+1}{2}} = 2^{\frac{m}{2}}\sigma^m \left(\frac{m-1}{2}\right) \left(\frac{m-3}{2}\right)\cdots \left(\frac{3}{2}\right) \left(\frac{1}{2}\right)$$
This equality is given in my lecture notes. What I can't understand is where the $\ \dfrac{1}{\sqrt{\pi}}\ $ went?
$$\Gamma\left(\frac{m+1}{2}\right) = \frac{m-1}{2}\cdot\frac{m-3}{2}\cdot\dotsc\cdot\frac{1}{2}\cdot\Gamma\left(\frac{1}{2}\right),$$
and $\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}$. So it cancelled against $\Gamma\left(\frac{1}{2}\right)$.