# Sharp upper bounds on hypergeometric function ${}_2F_1[a,b,c;z]$ when $|z|\geq1$

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.

The Gauss hypergeometric function satisfies the following functional equation: $${}_2F_1(a,b;c;z) = (1-z)^{-a}{}_2F_1\left(a,c-b,c;\frac{z}{z-1}\right).$$ So, you can write your hypergeometric function as \begin{align} {}_2F_1\left(-y,-j;1+x-j;-\frac{\gamma}{1-\gamma}\right) &= \left(1+\frac{\gamma}{1-\gamma}\right)^{y}{}_2F_1\left(-y,1+x;1+x-j;\frac{-\frac{\gamma}{1-\gamma}}{-\frac{\gamma}{1-\gamma}-1}\right)\\ &= (1-\gamma)^{-y}{}_2F_1\left(-y,1+x;1+x-j;\gamma\right). \end{align} This should allow you to use the bounds you already have for $0<z<1$.