Is there any way of writing the following contour integral as a closed form expression ? $$ \int_{-i \infty}^{+i \infty} dp \int_{-i \infty}^{+i \infty} ds \Gamma(d/2 + s + p -1) \Gamma(-\Delta - s - 2p +1)\Gamma(p) \Gamma(s+\Delta) \Gamma(-s) x^s y^p $$
where $\Delta \in \mathbb{C}$, $d \in \mathbb{N^*}$, and $x$ and $y$ are two complex independent variables.
I managed to write in a closed-form one of the two integrals, using the integral representation of the Gaussian hypergeometric function https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/07/02/ (after a change of variable $s \rightarrow -s$) but then I am still left with
$$ 2\pi i \Gamma(\Delta)\int_{-i \infty}^{+i \infty} dp \frac{\Gamma(d/2 + p -1) \Gamma(d/2 - p - \Delta)\Gamma(1 - 2p)}{\Gamma(d/2 - p)} y^p \space _2F_1(\Delta, d/2 + p -1, d/2 - p, 1-x) $$
which I don't really know how to write in a closed form.