I need to evaluate a residue and this term: $$_2F_1\left(\frac{\Delta}{2} - \frac{1}{8},\frac{\Delta}{2} + \frac{7}{8};\Delta;1\right),$$ which appears inside that expression. This makes the residue divergent. I checked using mathematica this gives complexinfinity independent of the value of $\Delta$.
I want to know how this term diverges. Tried to apply several available identities without any success.
Edit : Similarly, $$_3F_2\left(\frac{\Delta }{2}-\frac{9}{8},\frac{\Delta }{2}+\frac{7}{8},\frac{\Delta }{2}+\frac{7}{8};\frac{\Delta }{2}-\frac{1}{8},\Delta ;1\right)$$ diverges for any value of $\Delta$. There should be some identity that's applicable to any $\, _{p+1}F_p$ function, it seems.
The fundamental reason is that the hypergeometric differential equation for $_2F_1(a,b,c,z)$ is characterized by the Riemann symbol $$\left\{ \begin{array}{ccc}0 & 1 & \infty \\ 0 & 0 & a \\ 1-c & c-a-b & b\end{array}\right\}.$$ This means, in particular, that (for sufficiently generic parameters) $_2F_1(a,b,c,z)$ can be represented as a linear combination of two functions of the form $$f(z),(1-z)^{c-a-b}g(z),$$ where $f(z)$, $g(z)$ are holomorphic in the neighborhood of $z=1$ and can be normalized as $f(0)=g(0)=1$. In your case $c-a-b=-1$, which produces divergence. Similar reasoning can also be applied to $_{p+1}F_p$.