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I have an instance of the Meijer G function (using the definition from, first equation there) that seems like, given its simplicity, it should be expressible in terms of another special function. The instance is: $G\left(z\Bigg|\begin{matrix}- & - \\ 0,\tfrac{1}{2},1 & -\end{matrix}\right)=\frac{1}{2\pi i}\int_{\gamma_L}ds\, z^{-s} \frac{1}{\Gamma(-s)\Gamma(\tfrac{1}{2}-s)\Gamma(1-s)}$.

Thanks in advance for your scholarly input. >wink<

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Help appreciated -- anybody? – MarkWayne Apr 26 '12 at 4:00
May I ask where this came from? – J. M. Apr 26 '12 at 11:24
J.M.: see comment below ying's answer. – MarkWayne May 14 '12 at 21:23

You can write this function as an abramowitz function, it may or may not be easier to deal with.

MeijerG[{{}, {}}, {{0, 1/2, 1}, {}}, x^2/4]/(2 Sqrt[pi]) = Int_0^inf c^1 Exp (-c^2-x/c) dc

Source: Evaluate this integral in Mathematica and you should get the corresponding Meijer G function.

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Thanks for this answer. In fact, I arrived at the Meijer G function via exactly just this integral expression. The answer to J.M.'s question follows from just this observation. The integral arises in the determination of the Maxwellian averaged cross section times the velocity for a cross section assumed to be of 'Gamow form.' In fact, I think this is irreducible. – MarkWayne May 14 '12 at 21:22

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