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On the properties of Laplace transform originating from one-sided Lévy stable laws

We consider the conventional Laplace transform of $f(x)$, denoted by $\mathcal{L}[f(x); p]~\equiv~F(p)=\int_{0}^{\infty} e^{-p x} f(x) dx$ with ${\rm \mathfrak{Re}}(p) > 0$. For $0 < α< 1$ we furnish the closed form expressions for the inverse Laplace transforms $\mathcal{L}^{-1}[F(p^α); x]$ and $\mathcal{L}^{-1}[p^{α-1}F(p^α); x]$. In both cases they involve definite integration with kernels which are appropriately rescaled one-sided Lévy stable probability distribution functions $g_α(x)$, $0 < α< 1$, $x > 0$. Since $g_α(x)$ are exactly and explicitly known for rational $α$, \textit{i.e.} for $α= l/k$ with $l, k=1, 2, \ldots$, $l < k$, our results extend the known and tabulated case of $α= 1/2$ to any rational $0 < α< 1$. We examine the integral kernels of this procedure as well as the resulting two kinds of Lévy integral transformations.