Given the following contour integral

$$\frac{1}{2\pi j}\int^{c+j\infty}_{c-j\infty} \frac{\Gamma(-1+a+s)\Gamma(b+s)}{\Gamma(3+a-s)}\cos(-1+a+s)\, {}_2F_1\Big(-1-a+s,-1+a+s;\frac{1}{2};z\Big) y^s\: \mathrm{d}s ,$$

where $a,b,z,y \in \mathbb{R}$, as noticed there are two poles given by

$$P^{(1)}_k = 1-a-k \quad P^{(2)}_k=-b-k \quad\text{ where }\quad k=0,1,2,\dotsc,\infty.$$

The questions are:

- What is the suitable contour to sum the residues and solve the integral?
- Does the zero caused by $\Gamma(3+a-s)$ cancel any of the poles?