I need to investigate the asymptotic behaviour of the following Meijer G function at the origin \begin{equation} G_{1,2}^{2,0}(x) \left(\begin{matrix} 1-a\\ 0, 0 \end{matrix} \bigg| z \right) = \frac{1}{2\pi i}\oint\limits_{L}\frac{\Gamma^2(-s)}{\Gamma(1-a-s)}z^sds, \label{eq1} \end{equation} where $a>1$ and is noninteger, $L$ is a loop beginning and ending at $+\infty$, encircling all poles of $\Gamma(−s)$ that are $k=0,1,...$
According to the residue theorem $$\frac{1}{2\pi i}\oint\limits_{L}\frac{\Gamma^2(-s)}{\Gamma(1-a-s)}z^sds = \sum_{k=0}^\infty Res\left(\frac{\Gamma^2(-s)}{\Gamma(1-a-s)}z^s, k\right),$$ where the integrand has double poles for all $k=0,1,2...$ Thus,
$$Res\left(\frac{\Gamma^2(-s)}{\Gamma(1-a-s)}z^s, k\right) = \lim_{s\to k}\frac{d}{dz}\left[ (s-k)^2\frac{\Gamma^2(-s)}{\Gamma(1-a-s)}z^s\right]$$ $$= \frac{z^k\ln z + \Phi(1-a-k)z^k + c_k\gamma z^k}{(k!)^2\Gamma(1-a-k)},$$ where $\Phi(\cdot)$ is a polygamma function, and $\gamma$ is the Euler–Mascheroni constant, and $c_k$ are some constants depending on $k$. The correctness of my calculations can be checked in Wolfram. Thus, from the definition of Meijer G and above calculations it follows that
$$ G_{1,2}^{2,0}(x) \left(\begin{matrix} 1-a\\ 0, 0 \end{matrix} \bigg| z \right) = \sum_{k=0}^\infty \frac{z^k\ln z + \Phi(1-a-k)z^k + c_k\gamma z^k}{(k!)^2\Gamma(1-a-k)}$$
$$= \ln z \sum_{k=0}^\infty \frac{z^k}{(k!)^2\Gamma(1-a-k)} + R(a,z) = \frac{\ln z \; {}_{1}F_{1}(a,1,-z)}{\Gamma(1-a)} + R(a,z),$$ where $R(a,z)\to const$ as $z\to 0.$ Thus,
$$G_{1,2}^{2,0}(x) \left(\begin{matrix} 1-a\\ 0, 0 \end{matrix} \bigg| z \right) \sim \frac{\ln z \; {}_{1}F_{1}(a,1,-z)}{\Gamma(1-a)}$$ as $z\to 0.$
However, when I run the following code in MatLab $2020$b, I get the following plot showing that my calculations are not correct.
syms a;
a = 2.5;
x = 0.01:0.01:0.1;
plot(x, meijerG([],[1-a], [0,0], [],x)./(log(x).*hypergeom(a,1,-x)./gamma(1-a)))
$\mathrm{MeilerG}/(\ln z * \mathrm{hypergeom}(a,1,-z))$:
I have reviewed other topics related to this problem Guidance on reducing this Meijer-G function, Asymptotic value of Meijer G-function and it looks like my approach to this integral should be correct. Please, let me know if I missed something in my calculations. Any help would be much appreciated.
Expanding as series around $z=0$ $$f(z)=G_{1,2}^{2,0}\left(z\left| \begin{array}{c} 1-a \\ 0,0 \\ \end{array} \right.\right)$$ $$f(z)=\frac{-\psi ^{(0)}(1-a)-\log (z)-2 \gamma }{\Gamma (1-a)}+\frac{ (-\psi ^{(0)}(-a)-\log (z)-2 \gamma +2)}{\Gamma (-a)}z+O\left(z^2\right)$$
$$g(z)=\frac{\log (z) \, _1F_1(a;1;-z)}{\Gamma (1-a)}$$ $$g(z)=\frac{\log (z)}{\Gamma (1-a)}-\frac{\log (z)}{\Gamma (-a)}z+O\left(z^2\right)$$
$$f(z)\color{red}{\mathbf{+}}g(z)=\frac{\psi ^{(0)}(1-a)+2 \gamma }{\Gamma (1-a)}-\frac{\psi ^{(0)}(-a)+2 \gamma -2}{\Gamma (-a)}z+O\left(z^2\right)$$