I know and can show that
\begin{equation*} {}_{1}F_{1}\left(1,s,z\right)=1+\frac{z}s\,{}_{1}F_{1}\left(1,s+1,z\right) \end{equation*}
From here, I get: \begin{equation*} \begin{split} {}_{1}F_{1}\left(1,s+1,z\right)&= \frac sz\left\{{}_{1}F_{1}\left(1,s,z\right)-1\right\} \\ & = \frac sz\left[\frac {s-1}z\left\{{}_{1}F_{1}\left(1,s-1,z\right)-1\right\}-1\right]\\ & = \frac{s!}{z^s}\exp{(z)} - \sum_{l=0}^{s-1} \frac{s!}{z^{s-l}l!}. \end{split} \end{equation*}
From the suggested reference to Eq. (8.5.1), this means that for integers,
\begin{equation*} \gamma(s,z) = =\frac{s!}{z^s}\left(1-\exp{(-z)}\sum_{l=0}^{s-1}\frac{z^l}{l!} \right) = \frac{\Gamma(s+1)}{z^s}\left(1-\exp{(-z)}\sum_{l=0}^{s-1}\frac{z^l}{\Gamma(l+1)} \right) \end{equation*}
My question is, does the right hand side also hold for non-integer $s$, but more importantly if there is further simplification possible for the expression of ${}_{1}F_{1}\left(1,s+1,z\right)$, or if there is some other easier representation fot this special case?
Thanks for any insight!