I am looking for: $$_{1}{{F}_{1}}\left( 1,c,(a+b)x \right)-\, _{1}{{F}_{1}}\left( 1,c,ax \right)=(?)$$
where $x$ is between $0$ and $T$ and $a,b,c>0$.
I know the asymptotic behavior as $x$ goes to $\infty$, but at this situation that will not help me.
I also know the following relation, but not sure how to use it. $$\frac{\partial _{1}{{F}_{1}}\left( 1,c,x \right)}{\partial x}=\frac{1}{c}\, _{1}{{F}_{1}}\left( 2,c+1,x \right)$$
Thanks for your help!