We know that for all $n\in\mathbb{N}$ and $a\in\mathbb{R}\setminus \{0\}$
$$ D^{n }e^{a x} = a^{n}e^{a x}$$
So I thought that the fractional derivative of this function would be
$$ D^{\alpha }e^{a x} = a^{\alpha}e^{a x}$$
for $\alpha\in\mathbb{R}$.
However, there are different definitions for fractional derivatives and with the typical fractional derivative (Liouville) this is not the case, it uses the incomplete Gamma fnction. So, I would like to know if there are any definition in which
$$ D^{\alpha }e^{a x} = a^{\alpha}e^{a x}$$
holds.
Thank you very much.