Tricomi's (confluent hypergeometric) function is $$U(a,b,z) = \frac{1}{\Gamma(a)}\int_0^\infty e^{-zt}t^{a-1}(1+t)^{b-a-1}dt$$ for $\Re(a)>0$.
Is there any analytical expression/special function for the integral over a finite domain $$\int_x^y e^{-zt}t^{a-1}(1+t)^{b-a-1}dt,$$ where $x,y\in\mathbb{C}$ are constants?
The answer was given by @Maxim in the comments. The special function is called Humbert series. With the integral representation here, a substitution $t=ux$ immediately implies
$$\int_0^x (1+t)^{b-a-1}t^{a-1}e^{-zt} \mathrm{d}t = \frac{x^a}{a} \Phi_1(a,a-b+1,a+1;-x,-zx)$$
because $\frac{\Gamma(a)\Gamma(1)}{\Gamma(a+1)}=\frac{1}{a}$.