Let me use the notation from Maple <http://www.maplesoft.com/support/help/Maple/view.aspx?path=MeijerG> for the Meijer G-function. Then let me define,


$f\_+(x) = MeijerG( [[+1/2],[]], [[0,0],[]], x )$


$f\_-(x) = MeijerG( [[-1/2],[]], [[0,0],[]], x )$


Then by numerical evaluation I was able to show that


$\lim\_{x\to 0} \frac{f\_-(x)}{f\_+(x)} = 1/2$


What would the further terms be in an expansion around $x=0$? I guess there will be all sorts of logarithms and other nasty singularities...


(No, Maple 12, which I have installed, is not able to do a series expansion. I don't have access to Mathematica so I don't know about that.)



Series representation of ratio of two Meijer G-functions
According to Maple 16, both $f\_+(x)$ and $f\_-(x)$ can be expressed in terms of BesselK:


$$\eqalign{f\_+(x) &= \sqrt {\pi }\;{{\rm e}^{x/2}}\;{K\_0 \left(x/2\right)}\cr
f\_-(x) &= \frac{\sqrt {\pi }}{2}{{\rm e}^{x/2}}\left( \left( 1+x \right) { K\_0\left(x/2\right)}-x\;
{ K\_1\left(x/2\right)} \right) 
\cr}$$


We then have the series (not a Taylor series, because of the logarithmic terms)


$$\frac{f\_-(x)}{f\_+(x)} = {\frac {\ln(x) - 2\;\ln \left( 2 \right) +\gamma+
2}{2 \; \ln(x) - 4\;\ln \left( 2 \right) +2\; \gamma}}+{\frac {x
}{2}}+O \left( {x}^{2} \right) 
$$


Maple can, of course, provide as many terms as desired, but they look complicated.
The coefficient of $x^n$ for odd integers $n > 1$ appears to be $0$, while for even integers $n$ it appears to be of the form $C\_n(\ln(x))/(\ln(x) - 2 \ln(2) + \gamma)^{1+n/2}$ where
$C\_n$ is a polynomial of degree $1+n/2$. Thus if $v = \ln(x)-2 \ln(2) + \gamma$, the 
series can be written as 


$$\eqalign{
&{\frac {v+2}{2v}}+{\frac {x}{2}}+{\frac {2{v}^{2}-2v+1
}{16{v}^{2}}}{x}^{2}-{\frac {8{v}^{3}-20{v}^{2}+
21v-8}{2048{v}^{3}}}{x}^{4}\cr &+{\frac {72{v}^{4}-276
{v}^{3}+451{v}^{2}-351v+108}{442368{v}^{4}}}{x}^{6}\cr&-{\frac {3168{v}^{5}-16248{v}^{4}+36383\,{v}^{3}-43148
\,{v}^{2}+26784v-6912}{452984832{v}^{5}}}{x}^{8}\cr&+{
\frac {136800{v}^{6}-877560{v}^{5}+2507131{v}^{4}-4011375{v}^{
3}+3758250{v}^{2}-1944000v+432000}{452984832000{v}^{6}}}{x}^{10}\cr &+O \left( {x}^
{12} \right) \cr}
$$


