Asymptotic analysis of an expression involving a Fox's H function One of the performance metrics calculated in the analysis of telecommunications systems is the ergodic channel capacity, $C_{\rm erg}$. During one of my studies, I found the expression below for such a metric considering a Rayleigh channel and AWGN.
\begin{align}\label{eq:CAPACIDADEERGODICA}
     C_{\rm erg} & =  
       \frac{ z^2 }{2 \bar{\gamma} h_{\rm l}^{2}A_{0}^{2}\log(2)} {\rm H}_{3,4}^{4,1}\left[ \frac{1}{\bar{\gamma}h_{\rm l}^{2}A_{0}^2} \ \middle\vert \begin{array}{c}
     (-1, 1), (\frac{z^2}{2}, 1), (0, 1)  \\
     (0, 1), (\frac{z^2}{2}-1, 1), (-1, 1), (-1, 1)
       \end{array}\right]
\end{align}
To provide more insights into high SNR regimes ($\bar{\gamma} \to \infty$), the asymptotic behavior of this metric is often required.
Therefore, I'd like to know the asymptotic expression of the ergodic capacity, given by the above equation when $\bar{\gamma} \to \infty$.
OBS.: All the other variables can be considered constant.
UPDATE 10/01/2023
Additional information on how the previous equation is derived.
The ergodic capacity is found as the expectation of $\log_2(1+\gamma)$:
\begin{equation}\label{eq:Capacidade}
    C_{\rm erg} = \int_{0}^{\infty}\log_2(1+\gamma)f_{\Gamma}(\gamma){\rm d}\gamma,
\end{equation}
where
\begin{align}\label{eq:PDFsnr}
     f_{\Gamma}(\gamma) & =  
       \frac{ z^2}{2 \bar{\gamma} h_{\rm l}^{2}A_{0}^{2}} {\rm H}_{1,2}^{2,0}\left[ \frac{\gamma}{\bar{\gamma} h_{\rm l}^{2}A_{0}^2} \ \middle\vert \begin{array}{c}
     (\frac{z^2}{2},1)  \\
     (0, 1), (\frac{z^2}{2}-1,1)
       \end{array}\right]
\end{align}
is the PDF of the instantaneous SNR.
 A: Your expression can be simplified as follows: As $m=q=4$, the denominator parameter $(0,1)$ cancels with the numerator parameter $(0,1)$. Furthermore, as all linear coefficients are one, the Fox H function reduces to the simpler Meijer G function. I'll denote
$$\tag{1}
\xi = \frac 1 {\bar\gamma h_1^2 A_0^2},\quad \zeta = \frac{z^2}{2},
$$
such that
\begin{align}\tag{2a}
     C_{\rm erg} & =  
       \frac{ \zeta \xi}{\log 2} {\rm G}_{2,3}^{3,1}\left[ \xi \, \middle\vert \begin{array}{c}
     -1, \zeta \\
     -1, -1, \zeta-1
       \end{array}\right] \\
&= \frac{\zeta}{\log 2} {\rm G}_{2,3}^{3,1}\left[ \xi \, \middle\vert \begin{array}{c}
     0, \zeta+1 \\
     0, 0, \zeta
       \end{array}\right]\tag{2b}\\
&= \frac{1}{\log 2}\left( 
\frac {\pi \, \Gamma(1-\zeta)} {\sin(\pi\zeta)} \xi^\zeta
-\gamma_\mathrm E - \zeta^{-1} - \log \xi + \mathcal O(\xi)
\right),\tag{2c}
\end{align}
with Euler's constant $\gamma_\mathrm E$, the Gamma function $\Gamma$, and with the usual big $\mathcal O$ notation. The calculation was done in Mathematica.
