I'm currently going through this paper and I've run into an issue with a contour integral.
The relevant part of section 2.2 is quoted:
We now wish to study the analytic structure of (37). In order to do that, it is convenient to use the following Mellin-Barnes representation
$$ M(s) = \frac{-1}{\Gamma\left(\frac{\Sigma_i\Delta_i-d}{2}\right)\Gamma\left(\frac{\Delta_1+\Delta_3-s}{2}\right)\Gamma\left(\frac{\Delta_2+\Delta_4-s}{2}\right)}\int_{-i\infty}^{i\infty} \frac{dc}{2\pi i} \frac{l(c)l(-c)}{\left(c-\Delta+\frac{d}{2}\right)\left(c+\Delta-\frac{d}{2}\right)} \tag{38} $$
where
$$ l(c) = \frac{\Gamma\left(\frac{\frac{d}{2}+c-s}{2}\right)\Gamma\left(\frac{\Delta_1+\Delta_3+c-\frac{d}{2}}{2}\right)\Gamma\left(\frac{\Delta_2+\Delta_4+c-\frac{d}{2}}{2}\right)}{2\Gamma(c)} \tag{39} $$
Poles in $s$ arise from pinching of the integration contour in (38) between two colliding poles of the integrand. In fact, we can write
$$ M(s) = -\sum_{n=0}^\infty \frac{R_n}{s-\Delta-2n} \tag{40} $$
with
$$ R_n = \frac{\Gamma\left(\frac{\Delta_1+\Delta_3+\Delta-d}{2}\right)\Gamma\left(\frac{\Delta_2+\Delta_4+\Delta-d}{2}\right)\left(1+\frac{\Delta-\Delta_1-\Delta_3}{2}\right)_n\left(1+\frac{\Delta-\Delta_2-\Delta_4}{2}\right)_n}{2n!\Gamma\left(\frac{\Sigma_i\Delta_i-d}{2}\right)\Gamma\left(\Delta-\frac{d}{2}+n+1\right)} \tag{41} $$
$\Delta,\Delta_i$, and $d$ are fixed but arbitrary, the integration contour is parallel to the imaginary axis, and $(x)_n$ denotes the Pochhammer symbol/rising factorial.
I can see that as $s\to \frac{d}{2}+2n$ for $n\in\mathbb{N}$, for each $n$ a pair of poles from $\Gamma\left(\frac{\frac{d}{2}+c-s}{2}\right)\Gamma\left(\frac{\frac{d}{2}-c-s}{2}\right)$ will collide at $c=0$, leaving the integration poorly defined and requiring some sort of contour deformation.
My issue is in reproducing equation (41). If we deform the contour to pick up the poles at $c=s-\frac{d}{2}-2n$ then we indeed find an expression of the form
$$ \sum_{n=0}^\infty \frac{f(s,\Delta,\Delta_i,n)}{s-\Delta-2n} $$
but the numerator in each term has an $s$-dependence, while those in (41) do not. Trying a few examples numerically also indicates that the two expressions are not the same.
It's not clear to me how I should deform the contour; the usual methods for moving poles off a contour don't seem applicable here.
I'm clearly missing something (probably something obvious), and any help evaluating this integral would be appreciated.