Is there any way to simplify $U(a,a+1/2,z)$ or relate it to any other common special functions such as the incomplete beta function or the incomplete gamma function?
Here, $U(a,b,z)$ is Kummer's U function:
$U(a,b,z) = \frac{1}{\Gamma(a)}\int_{0}^{\infty}{ e^{-zt}t^{a-1}(1 + t)^{b-a-1}dt}$
I don't know if there is something analagous to $U(a,a+1,z) = z^{-a}$ or to $U(a,a,z) = e^{z}\Gamma(1-a,z)$, where $\Gamma(1-a,z)$ is the incomplete gamma function.
Also, I am assuming that both a and z are positive and real.