In a statistical physics book, I don't understand how they moved from this expression:
$\Big(\frac{N}{2}-m\Big)! \Big(\frac{N}{2}+m\Big)! = \Big(\frac{N}{2}!\Big)^2$
with $N=2k, k,m\in \mathbb{Z^+}$, to this:
$\Big(\frac{N}{2}+m\Big)\Big(\frac{N}{2}+m-1\Big)...\Big(\frac{N}{2}+1\Big) = \Big(\frac{N}{2}-m+1\Big)\Big(\frac{N}{2}-m+2\Big)...\Big(\frac{N}{2}\Big)$
I would appreciate any help thanks!
For any integer $a$, you can write \begin{align*} \require{cancel} (a-m)!(a+m)! &= (a!)^2 \\ \prod_{k=1}^{a-m} k \prod_{k=1}^{a+m} k &= \prod_{k=1}^a k^2 \\ \cancel{\left(\prod_{k=1}^{a-m} k\right)^2} \prod_{k=a-m+1}^{a+m} k &= \cancel{\left(\prod_{k=1}^{a-m} k\right)^2} \left(\prod_{k=a-m+1}^{a} k\right)^2 \\ \cancel{\prod_{k=a-m+1}^{a} k} \prod_{k=a+1}^{a+m} k &= \left(\prod_{k=a-m+1}^{a} k\right)^{\cancel{2}} \\ \prod_{k=a+1}^{a+m} k &= \prod_{k=a-m+1}^{a} k \end{align*} which is exactly what they have written with $a=N/2$. Sorry my notation is a bit messy... but it's just a bunch of cancellations.