On the properties of Laplace transform originating from one-sided L\'evy stable laws

We consider the conventional Laplace transform of $f(x)$, denoted by $\mathcal{L}[f(x);p]\equiv F(p)={\int }_0^{\mathrm{\infty }}{e}^{-px}f(x)dx$ with $\mathfrak{R}\mathfrak{e}(p)>0$. For $0<\alpha <1$ we furnish the closed form expressions for the inverse Laplace transforms $$\mathcal{L}$^{-1}[F(p^{\alpha}); x]$and$$\mathcal{L}$^{-1}[p^{\alpha-1}F(p^{\alpha}); x]$. In both cases they involve definite integration with kernels which are appropriately rescaled one-sided L'{e}vy stable probability distribution functions $g_{\alpha }(x)$, $0<\alpha <1$, $x>0$. Since $g_{\alpha }(x)$ are exactly and explicitly known for rational $\alpha$, $\mathit{\text{i.e.}}$ for $\alpha =l\mathrm{/}k$ with $l, k=1, 2, \ldots$,$l < k$,ourresultsextendtheknownandtabulatedcaseof$\alpha = 1/2$toanyrational$0 < \alpha < 1$. We examine the integral kernels of this procedure as well as the resulting two kinds of L'{e}vy integral transformations.