If I have two positive functions $f,\ g$ supported on $x>0$. Also, If I have: $$\mathcal{L}\{f\}(s)=F(s)$$ and $$\mathcal{L}\{g\}(s)=F(s)^r$$ Where $\mathcal{L}$ is the Laplace transform and $r\in \mathbb{R}_+$.
What can I say about $g$ ?
If $r\in \mathbb{N}$ then by the uniqueness of the Laplace transform $g=\underbrace{f*f*\dots*f}_{r}$.
Now if $r\in\mathbb{R}_+$ then If we take $\Omega(F)=F^r$we can use Taylor expansion around the constant function 1 to get: $$\sum c_k(F-1)^k=F^r$$ $$\implies \sum{(r)_k(F-1)^k\over k!}=F^r$$ Taking the inverse laplace we get: $$g=\sum {(r)_k(f-\delta)^{*k}\over k!}$$ Is this correct?