Using:
$$\sum _{j=0}^{\infty } -s^{(1+j) (-1-\alpha )} (s-\lambda )^j=-\frac{1}{-s+s^{1+\alpha }+\lambda }$$
then we have:
Sum[FullSimplify[
InverseLaplaceTransform[-s^((1 +
j) (-1 - \[Alpha])) (s - \[Lambda])^j, s, t],
Assumptions -> {t > 0, \[Lambda] > 0, \[Alpha] > 0}], {j, 0,
Infinity}]
$$\sum _{j=0}^{\infty } -t^{(1+j) \alpha } \, _1\tilde{F}_1(-j;1+\alpha +j \alpha ;t \lambda )$$
I doubt there's a closed form for the series.
Regards M.I.