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?
$$_{2}F_{2}\left(a,b;c,d;z\right)=\frac{\Gamma (c) \Gamma (d)}{\Gamma (a) \Gamma (b)}e^z z^{a+b-c-d}\left(1+{\cal O}(z^{-1})\right)$$
As an example, the plot shows $_{2}F_{2}\left(a,b;c,d;z\right)$ (blue) and the asymptotics (gold) for $a=1,b=2,c=3,d=4$.