How can we transfer $\frac{1}{{1 + \delta x}}G_{0,1}^{1,0}\left[ {\theta x\left| {_n^ - } \right.} \right]$ into the form like $G_{u,v}^{m,n}\left( {} \right)$, where $\sigma$, $x$, and $\theta$ are positive real numbers, and $n$ is a positive integer. The MeijerG function can be found in https://en.wikipedia.org/wiki/Meijer_G-function
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I guess it is not possible to convert into a single MeijerG. But you can convert it into product of two MeijeerG then into a single MeijerG (MeijerG[{{1, 1}, {}}, {{1}, {0}}, z])
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