# Series representation of ratio of two Meijer G-functions

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.)

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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}

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It turns out the only reason maple 12 couldn't do the expansion is the I use Fedora 16 linux distribution and there is a known bug in maple that only happens on this platform: mapleprimes.com/questions/… –  Daniel Oct 26 '12 at 21:03