## 3 Statistical Characteristics of the Sum of \(\alpha\)-\(\mu\)/IGA RVs
### Approximated Statistics of the Sum of the Independent RVs  
By using (A1) and (12), the approximated MGF of \(\gamma_{MRC}\) can be obtained as  
\[MGF_{{}_{\mathcal{M}RC}}(s)=\prod\limits_{l=1}^{L}MGF_{{}_{\gamma_{l}}}(s)= \prod\limits_{l=1}^{L}\Bigg{(}\sum\limits_{i=1}^{N_{l}}\frac{\xi_{i}\alpha_{ l}}{2\Gamma(\mu_{l})}H_{1,1}^{1,1}[(s\psi_{i_{l}})^{-0.5\alpha_{l}}|_{(\mu_{l},1)} ^{(1,0.5\alpha_{l})}]\Bigg{)}. \tag{22}\]  
For ease of calculation, we swap the order of the sum and the product in the right-hand side formula in (22); (22) can then be re-expressed in multiple summation forms as  
\[MGF_{{}_{\mathcal{M}RC}}(s)=\sum\limits_{i_{1}=1}^{N_{1}}\cdots\sum\limits_{i _{L}=1}^{N_{L}}\prod\limits_{l=1}^{L}\Bigg{(}\frac{\xi_{i_{l}}\alpha_{l}}{2 \Gamma(\mu_{l})}H_{1,1}^{1,1}[(s\psi_{i_{l}})^{-0.5\alpha_{l}}|_{(\mu_{l},1)} ^{(1,0.5\alpha_{l})}]\Bigg{)} \tag{23}\]  
By utilizing (23) and the similar derivation procedure as Appendix A, the approximated PDF of \(\gamma_{MRC}\) can be obtained as  
\[f_{{}_{\mathcal{M}RC}}(\gamma)=\sum\limits_{i_{1}=1}^{N_{1}}\cdots\sum\limits_ {i_{L}=1}^{N_{L}}\Bigg{[}\Bigg{(}\prod\limits_{l=1}^{L}\frac{\xi_{i_{l}}\alpha _{l}}{2\Gamma(\mu_{l})}\Bigg{)}\gamma^{-1}H_{0,1;[1,1]_{l=L}}^{0,0;[1,1]_{l=L} }[\left(\gamma/\psi_{i_{1}}\right)^{0.5\alpha_{1}},\cdots,\left(\gamma/\psi_{i _{L}}\right)^{0.5\alpha_{L}}|_{(1;0.5\alpha_{l})_{l=1:L}}^{-}|_{([1,0.5\alpha_ {l})_{l})_{l=1:L}}|_{([\mu_{l},1)]_{l=1:L}}^{[(1,0.5\alpha_{l})]_{l=1:L}}|_{([ \mu_{l},1)]_{l=1:L}}^{[(1,0.5\alpha_{l})]_{l=1:L}}\Bigg{]}\Bigg{]}. \tag{24}\]  
From (21) and (24), we can see the summation forms in (24) bring the reduction of arguments in the multivariate Fox's H-function. Importantly, (24) is a unified analytical expression and can be applied to more channel models where the PDF can be expressed as a mixture \(\alpha\)-\(\mu\) form in (10), such as \(\alpha\)-\(\eta\)-\(\mu\) and \(\alpha\)-\(\kappa\)-\(\mu\), \(\alpha\)-\(\eta\)-\(\mu\)/lognormal and \(\alpha\)-\(\kappa\)-\(\mu\)/lognormal in [5], \(\alpha\)-\(\mu\)/gamma in [6], and \(\alpha\)-\(\mu\)/\(\alpha\)-\(\mu\) in [12]. Hence, the corresponding approximated CDF of \(\gamma_{MRC}\) can be derived as  
\[F_{{}_{\mathcal{M}RC}}(\gamma)=\sum\limits_{i_{1}=1}^{N_{1}}\cdots\sum\limits_ {i_{L}=1}^{N_{L}}\Bigg{[}\Bigg{(}\prod\limits_{l=1}^{L}\frac{\xi_{i}\alpha_{l}}{2 \Gamma(\mu_{l})}H_{0,1;[1,1]_{l=L}}^{0,0;[1,1]_{l=1:L}}[\left(\gamma/\psi_{ i_{l}}\right)^{0.5\alpha_{1}},\cdots,\left(\gamma/\psi_{i_{L}}\right)^{0.5\alpha_{L}}|_{(0; 0.5\alpha_{l})_{l=1:L}}^{-}|_{([1,0.5\alpha_{l})_{l})_{l=1:L}}^{[(1,0.5\alpha_ {l})]_{l=1:L}}|^{[(1,0.5\alpha_{l})]_{l=1:L}}|^{[(1,0.5\alpha_{l})]_{l=1:L}}|^{ [(1,0.5\alpha_{l})]_{l=1:L}}|\Bigg{]}. \tag{25}\]  
In the i.i.d. case, (22) can be simplified as  
\[MGF_{{}_{\mathcal{M}RC}}(s)=\Bigg{(}\sum\limits_{i=1}^{N}\frac{\xi_{i}\alpha}{2 \Gamma(\mu)}H_{1,1}^{1,1}[(s\psi_{i})^{-0.5\alpha_{l}}|_{(\mu,1)}^{(1,0.5 \alpha_{l})}]\Bigg{)}^{L} \tag{26}\]  
By applying the multinomial theorem, (26) can be rewritten as  
\[MGF_{{}_{\mathcal{M}RC}}(s)=\sum\limits_{k_{1}+\cdots+k_{N}=L}\Bigg{(} \begin{array} []{c}L\\ k_{1},\cdots,k_{N}\end{array} \Bigg{)}\underbrace{\prod\limits_{1\leq i\leq N} \Bigg{(}\frac{\xi_{i}\alpha}{2\Gamma(\mu)}H_{1,1}^{1,1}[(s\psi_{i})^{-0.5\alpha }|_{(\mu,1)}^{(1,0.5\alpha_{l})}]\Bigg{)}^{k_{i}}}_{I_{1}}, \tag{27}\]where \(\sum\limits_{k_{1}+\cdots+k_{N}=L}=\sum\limits_{k_{1}=0}^{L}\sum\limits_{k_{2}=0}^ {L-k_{1}}\cdots\sum\limits_{k_{N-1}=0}^{L-k_{1}-\cdots-k_{N-2}}\), \(\left( \begin{array}{c}L\\ k_{1},\cdots,k_{N}\end{array} \right)=\frac{L! }{k_{1}!k_{2}!\cdots k_{N}!}\).