Can you please help me devise a series for the Meijer's G-function (i) with inceces m=3, n=0, p=1 and q=3, for a general real variable?
The first difficulty that I am facing is the proper choice of an integration path, to use the residues' theorem. It seems to me that I may choose between two possibilities, one of which encloses no pole, whereas the other contour does. That is a nonsense, of course, but I am failing to see where my errors lies...
(i) The single a-coefficient is 0, and the three b-coefficients are z, 0, 0 (z is a general complex number).
We have $$G_{1,3}^{3,0}\left( x \middle| {0 \atop z, 0, 0} \right) = \frac 1 {2 \pi i} \int_{\mathcal L} \frac {\Gamma(z-y) \Gamma(-y)^2} {\Gamma(-y)} x^y dy.$$ Notice that a pair of the gamma functions cancels out.
To determine the integration contour, you need to analyze the asymptotic behavior of the gamma function and choose a contour (separating the poles) over which the integral converges, as this is implied in the definition of the G-function. In this case only the right loop will do.
Next, writing the integral as a sum of residues, you get a series expansion for small $x$. The integrand has residues at the points $y=k$ and $y=z+k$, thus the leading term of the expansion may be of order zero, or of order $z$, or there may be a logarithmic case.