N. M. Temme, "Special Functions" (Wiley 1996) gives the following expression that expresses the upper incomplete gamma function in terms of the ordinary gamma function, for integer orders:
$$ \Gamma(n,z) = \Gamma(n) e^{-z} \sum_{m=0}^{n-1} \frac{z^m}{m!} \\ n=1,2,... $$
My experiments indicate that this is a convenient way to compute the upper incomplete gamma function for small integer orders as the computation appears to be numerically stable. I tried orders up to n=50 and a wide range of real z.
Is there a similar expression that allows the straightforward computation of the upper incomplete gamma function in terms of the ordinary gamma function, for half-integer orders, that is, $ \Gamma(n+\frac{1}{2},z) $? I am aware that $ \Gamma(\frac{1}{2},z) = \sqrt{\pi} erfc(\sqrt{z}) $.
According to Maple, for nonnegative integers $n$ $$ \Gamma(1/2+n, t) = \text{pochhammer}(1/2,n) \sqrt{\pi}\; \text{erfc}(\sqrt{t}) + t^{n-1/2} e^{-t} \sum_{k=0}^{n-1} \text{pochhammer}(1/2-n,k) (-t)^{-k} $$
Note that $\text{pochhammer}(1/2,k) = 2^{1-2k} \dfrac{(2k-1)!}{(k-1)!}$.