Roots of the incomplete gamma function Is there any way that one can describe all the roots of the incomplete gamma function $\Gamma(n,z)$, for $n\in \mathbb{N}$, analytically?

With the usual definition we have :
$$\Gamma(n,x)=\int_x^{\infty} t^{n-1} e^{-t} dt$$
After repeatedly using integration by parts we may get this formula :
$$\Gamma(n,x)= (n-1)! e^{-x} \sum_{k=0}^{n-1} \frac{x^k}{k!}$$
so that you are asking for the zeros of the polynomials : 
$$P_n(x)=\sum_{k=0}^{n-1} \frac{x^k}{k!}$$
Let's search the first solutions :
$
\begin{array} {rcc}
n & P_n(x) & \mathrm{zeros} \\
\hline \\
1 & 1 & \emptyset \\
2 & 1+x & \{-1\} \\
3 & 1+x+\frac{x^2}{2!} & \{-1-i,-1+i\} \\
4 & 1+x+\frac{x^2}{2!}+\frac{x^3}{3!} & \{\sqrt[3]{\sqrt{2}-1}-\frac1{\sqrt[3]{\sqrt{2}-1}}-1,..,.. \} \\
\end{array}
$
I'll stop here because the algebraic expressions of the zeros are becoming rather long and complicated... (search others with WolframAlpha)
Or did you hope something simpler?
