I am attempting to compute the (Kummer’s) confluent hypergeometric function (see also here)

\begin{align} {}_1F_1\left(\frac{n}{2};n +\frac{3}{2};z\right) \end{align}

for real $ z\lt0$ and many consecutive values of integer $ n\geq0$ . Since the first two arguments scale with $ n$ at different rates, the usual recurrence relations don’t apply. Additionally, it does not seem to fit into any of the usual special cases that relate $ {}_1F_1$ to other functions. Using Mathematica, we find that the first 7 terms are:

\begin{align} n&=0 &&1,\ n&=1 &&\frac{3 \sqrt{\pi } (2 z-1) \text{erf}\left(\sqrt{z}\right)}{8 z^{3/2}}+\frac{3 e^{-z}}{4 z},\ n&=2 &&\frac{15 \sqrt{\pi } e^{-z} \text{erfi}\left(\sqrt{z}\right)}{8 z^{5/2}}+\frac{5 (2 z-3)}{4 z^2},\ n&=3 &&\frac{105 \sqrt{\pi } (4 (z-3) z+15) \text{erf}\left(\sqrt{z}\right)}{128 z^{7/2}}+\frac{105 e^{-z} (2 z-15)}{64 z^3},\ n&=4 &&-\frac{945 \sqrt{\pi } e^{-z} (2 z+7) \text{erfi}\left(\sqrt{z}\right)}{64 z^{9/2}} + \frac{63 (8 (z-5) z+105)}{32 z^4},\ n&=5 &&\frac{3465 \sqrt{\pi } \left(8 z^3-60 z^2+210 z-315\right) \text{erf}\left(\sqrt{z}\right)}{1024 z^{11/2}}+\frac{3465 e^{-z} \left(4 z^2+315\right)}{512 z^5},\ n&=6 &&\frac{135135 \sqrt{\pi } e^{-z} (4 z (z+9)+99) \text{erfi}\left(\sqrt{z}\right)}{1024 z^{13/2}}+\frac{1287 (2 z (16 z (2 z-21)+1575)-10395)}{512 z^6} \end{align}

I would appreciate any tips/intuition for how to simplify this function.