I am trying to calculate the multiplication $e^{x} \, _2 F_2(a+1,a+1;a+2,a+2;-x)$, where $a>0$, and $x \in (a-\sqrt{a},a+\sqrt{a})$ approximately. But this expression is not calculable for large $a > 1000$.
I know there is a similar transformation $e^{x} \, _2 F_2(b-a-1,f+1;b,f;-x) = \, _2 F_2(a,c+1;b,c;x)$ sciencedirect.com/science/article/pii/S0377042704002237. Unfortunately, it is not what I need.
So, is it possible to relate the multiplication $e^{x} \, _2 F_2(a+1,a+1;a+2,a+2;-x)$ to some $\, _2 F_2(m,n;p,q;x)$? Or any suggestions on analytic approximations of this multiplication? Thanks.
If $a$ is a positive integer, the hypergeometric function can be expressed in "closed form". But the exponential factor is going to result in your expression growing rapidly (like $c\exp(x) \ln(x)/x^{a+1}$ I think).