Let $\gamma(n, u):=\int_u^\infty e^{-t} t^{n-1} dt$, where $u\in \mathbb R^+$ and $n\in \mathbb N$. This is the lower incomplete gamma function on a restricted domain. For a given $u$, is it possible to express analytically a function $F_u$ such that $$\left. \frac{d^n}{dx^n} F_u(x)\right|_{x=0}= \gamma(n, u)$$
or, such that $$F_u(x)=\sum_{n=0}^\infty \frac{\gamma(n, u)}{n!}x^n$$
Pretty straightforward. The series converges if $|x|<1$, and then, doing $\sum\int\mapsto\int\sum$, $$F_u(x)=\sum_{n=0}^\infty\frac{x^n}{n!}\int_u^\infty e^{-t}t^{n-1}dt=\int_u^\infty\frac{e^{(x-1)t}}{t}dt=E_1\left(\frac{u}{1-x}\right)$$ using the function $E_1(z)=\int_z^\infty\frac{e^{-t}}{t}dt$ closely related to the exponential integral.