I have to deal a lot with functions which are $_3F_2$ with unit argument, and I need to find their behavior for large values of the parameters. One example is
$$_3F_2(n,n,n;2n,n+x;1)$$
where I'm interested in the behavior of large $n$. Now, I have played around with this function numerically a bit and it looks to me that asymptotically (for $n \to \infty$) it behaves as
$$\frac{1}{2\sqrt{\pi}}4^n n^{1/2-x} \Gamma(x)$$
However, this is of course very dirty and I would like to have a clean way to get this kind of asymptotics. Is there a way to do this or are there some references with similar results?
The formula is correct if $\pi$ is changed to $\sqrt \pi$. For this particular example, we have $${_3\hspace{-1px}F_2}(n, n, n; 2 n, n + x; 1) = \frac {\Gamma(x) \Gamma(2 n)} {\Gamma(n) \Gamma(n + x)} \,{_3\hspace{-1px}F_2}(n, x, x; n + x, n + x; 1).$$ Now the $k$th term in the hypergeometric sum is of order $n^{-k}$. For $n \to \infty$ and $x$ fixed, $${_3\hspace{-1px}F_2}(n, x, x; n + x, n + x; 1) \sim 1, \\ \frac {\Gamma(x) \Gamma(2 n)} {\Gamma(n) \Gamma(n + x)} \sim \frac {\Gamma(x) n^{1/2 - x} 2^{2 n - 1}} {\sqrt \pi}.$$