I can't seem to nail the closed form of the incomplete upper gamma function for the number of degrees of freedom $s$ being a fraction type $n/2$ where $n$ is integer. For the case when $s$ is integer it's simple, wiki has the closed form. [closed form upper incomplete gamma for integer $s$][1]
It should be simple, as we also have the value for the $\Gamma(0.5, x)$ as the starting point, but I seem to arrive to a wrong result:
$$ \Gamma(s,x) = e^{-x}s! \left(\frac{\operatorname{erfc}(\sqrt x)e^{-x}\sqrt \pi}{2} + \sqrt x + \sum_{k=0}^{s-2}\frac{x^{s-k}}{(s-k)!}\right). $$
Help? [1]: https://i.stack.imgur.com/DjGyl.png
