What is the inverse Laplace transform of $\frac{1}{1+ks^{\alpha}}$?

Question

What is the inverse Laplace transform of F(s)=$\frac{1}{1+ks^{\alpha}}$, $k$ is the constant and $\alpha$ is another constant. Especially when $\alpha$ is not an integer.

Answer 1

With CAS help:

$$\mathcal{L}_s^{-1}\left[\frac{1}{1+k s^{\alpha }}\right](t)=\sum _{j=0}^{\infty } \frac{(-1)^j k^{-1-j} t^{-1+\alpha +j \alpha }}{\Gamma (\alpha +j \alpha )}=\frac{t^{-1+\alpha } E_{\alpha ,\alpha }\left(-\frac{t^{\alpha }}{k}\right)}{k}$$

where: $E_{\alpha ,\alpha }\left(-\frac{t^{\alpha }}{k}\right)$ is generalized Mittag-Leffler function.

Generalization:

$$\mathcal{L}_s^{-1}\left[\frac{1}{a+k s^{\alpha }}\right](t)=\frac{t^{-1+\alpha } E_{\alpha ,\alpha }\left(-\frac{a t^{\alpha }}{k}\right)}{k}$$