I am new in this group, learning about Meijer G-Function, but I didn't understand, how could I change into $e^{x}$ into G-Function?
$$ e^{x} = G^{1, 0}_{0, 1}\left({\frac {{-}}{{0}}}\, \Big\vert\,^{}_{-x}\right)$$ where $G$ is the Meijer G function.
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Which definition of the function do you want to use ? – Jul 29 '19 at 12:00
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Write out the G-function as a contour integral according to its definition. Prove that the residue theorem applies and use the fact that $\operatorname{Res}_{s = k} \Gamma(-s) = (-1)^{k + 1}/k!$ for $k \in \mathbb N^0$. – Maxim Jul 29 '19 at 18:17