# Asymptotic behavior of $_3F_2$ at unit argument for large values of parameters

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, $${_3\hspace{-1px}F_2}(n, x, x; n + x, n + x; 1) \sim 1, \\ \frac {\Gamma(x) \Gamma(2 n)} {\Gamma(n) \Gamma(n + x)} \sim \frac {\Gamma(x) n^{1/2 - x} 2^{2 n - 1}} {\sqrt \pi}.$$