We need to expand the function $_2F_2(a+b x,1; 1+a+b x, b x; x)$ near $x=+\infty$. Where $a$ is complex, $b>1$.
When $x\to+\infty$, both the parameters and the variable goes to infinity, we can not used the expansion formulas only for large parameters or only for large variable.
Any comments and references are welcomed.
Update Because $${_1F_1}(a;c;x)=\lim_{b\to\infty}{_2F_1}(a,b;c;x/b)$$
We have $$\lim_{y\to\infty}{_2F_2}(a+b x,1; 1+a+b x, b y;y)=_2F_1(a+b x,1; 1+a+b x;1/b)$$
So the leading term seems to to be $ _1F_0(1; ;1/b)=b/(b-1)$