Generally, hypergeometric function ${}_2F_1[a,b,c;z]$ is defined using Gauss series ${}_2F_1[a,b,c;z]=\sum_{n=0}^\infty\frac{(a)_n(b)_n}{(c)_nn!}z^n$ with $|z|<1$, and there seems to be a lot of literature relating ${}_2F_1[a,b,c;z]$ to power mean when $z\in(0,1)$ (see this for a survey).

I am wondering what is known about upper bounds on hypergeometric function ${}_2F_1[a,b,c;z]$ when $|z|\geq1$ (provided the series converges). Does anyone have any pointers to the literature?

**Specifics of my problem**

I am trying to upper-bound the following summation (which is known to be a probability, so trivial upper bound is one):

$$S(x,y)=\sum_{i=0}^y\sum_{j=0}^x\frac{i!j!}{x!y!}\gamma^{x-j}(1-\gamma)^{y+j}\left(\binom{x}{j}{}_2F_1\left[-y,-j,1+x-j,-\frac{\gamma}{1-\gamma}\right]\right)^2$$

where $x$ and $y$ are non-negative integers and $\gamma\in(0,1)$. The hypergeometric function inside the summation converges, and, I believe an upper bound on it would be helpful in upper-bounding the entire expression. Any help would be appreciated.