I would like to find an asymptotic expansion for the hypergeometric function


$$
\_{2}F\_{2}\left(a,b;c,d;z\right),\quad a,b,c,d\in\mathbb{R}.
$$


The parameters are fixed. $z$ is real and $z\rightarrow +\infty$.


Could someone shed light on it?



Asymptotic expansion of hypergeometric 2F2
$$\_{2}F\_{2}\left(a,b;c,d;z\right)=\frac{\Gamma (c) \Gamma (d)}{\Gamma (a) \Gamma (b)}e^z z^{a+b-c-d}\left(1+{\cal O}(z^{-1})\right)$$


As an example, the plot shows $\_{2}F\_{2}\left(a,b;c,d;z\right)$ (blue) and the asymptotics (gold) for $a=1,b=2,c=3,d=4$.


![](https://ilorentz.org/beenakker/MO/PFQplot.png)


