When I was trying to evaluate the left hand side (LHS) of the following identity numerically, I noticed that Mathematica is able to simplify it; see the right hand side. However, I have no idea how to prove this identity, or to simplify LHS. Any help or reference would be appreciated.
$$\sum _{k=0}^n \left( \begin{array}{c} n \\ k \\ \end{array} \right)(-1)^k\frac{\Gamma(\gamma +n-k)}{\Gamma(\gamma )}\frac{\Gamma(\gamma +k)}{\Gamma(\gamma )}=\frac{2^n \pi ^{3/2} \text{Csc}(n \pi +\pi \gamma )}{\Gamma\left(\frac{1-n}{2}\right) \Gamma\left(\frac{1}{2} (2-n-2 \gamma )\right) \Gamma(\gamma )},$$ where $\eta,\gamma>0$ and $n \in \mathbb{N}$.
I believe RHS is not correct for all $n$ and $\gamma$. Because it is infinity when $n+\gamma \in \mathbb{N}$.