This article states that the Meijer G-Function satisfies a linear differential equation. The Gamma function is a special case of the incomplete Gamma function, which is a special case of the Meijer G-Function, so the Gamma function is the solution of a linear differential equation, contradicting Holder's Theorem.
Where is the mistake in this argument?
The mistake is that in specialising from the incomplete Gamma-function to the Gamma-function, you are putting equal to a constant the argument which is the one being differentiated in the Meijer $G$ differential equation.
To be more specific, for $\Re(a)>0$, the incomplete Gamma-function is $$ \gamma(a,z) = \int_0^z t^{a-1} e^{-t} \, dt = G_{1,2}^{1,1}\left( \begin{matrix}1 \\ a,1 \end{matrix} \middle| z \right) = a^{-1}z^a e^{-z} {}_1F_1(1;a+1;z) , $$ which satisfy a differential equation in the variable $z$, namely $$z y'' + (z-a+1) y' = 0 $$
But the ordinary Gamma-function is $\lim_{z \to \infty} \gamma(a,z)$ with the limit taken in the right half-plane. This does not depend on $z$, so the differential equation does not apply to it. There is no differential equation that involves the variable $a$, which is what would be required.