MB integrals and hypergeometic funcitons

Question

I'm trying to understand a step in ArXiv:[math-ph/1104.2661]. Equation 3.4 reads, \begin{equation} \frac{1}{\Gamma(2\epsilon)}\int_{-i\infty}^{i\infty}\frac{d\omega}{2i\pi}\frac{(-t)^\omega}{(-s)^{2-\epsilon+\omega}}\Gamma^2(\omega+1)\Gamma(2-\epsilon+\omega)\Gamma(-\omega)\Gamma^2(\epsilon-1-\omega) \end{equation} then, by taking the poles $\omega=\epsilon-2-n$ the result reads, \begin{equation} \frac{\Gamma(\epsilon)^2\Gamma(1-\epsilon)^2}{\Gamma(2\epsilon)\Gamma(2-\epsilon)}(-t)^{\epsilon-2}\ _2F_1(1,1,2-\epsilon,-\frac{s}{t}) \end{equation} The question is very simple. The residue theorem give the following result, \begin{equation} \sum_{n=0}^\infty \Gamma(\epsilon-1-n)^2\Gamma(n+2-\epsilon)(-1)^nn!\frac{(-t)^{\epsilon-2-n}}{(-s)^{-n}} \end{equation} The definition of the hypergeometic reads, \begin{equation} _2F_1(a,b,c,z)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\sum_{n=0}^\infty\frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}\frac{z^n}{n!} \end{equation} I don't see what I have to do in order to obtain the right answer.

This post imported from StackExchange Physics at 2014-08-09 08:48 (UCT), posted by SE-user user47354