On the sum of gamma random variates with application to the performance of maximal ratio combining over Nakagami-m fading channels

The probability distribution function (PDF) and cumulative density function of the sum of L independent but not necessarily identically distributed gamma variates, applicable to maximal ratio combining receiver outputs or in other words applicable to the performance analysis of diversity combining receivers operating over Nakagami-m fading channels, is presented in closed form in terms of Meijer G-function and Fox H̅-function for integer valued fading parameters and non-integer valued fading parameters, respectively. Further analysis, particularly on bit error rate via PDF-based approach, too is represented in closed form in terms of Meijer G-function and Fox H̅-function for integer-order fading parameters, and extended Fox H̅--function (Ĥ) for non-integer-order fading parameters. The proposed results complement previous results that are either evolved in closed-form, or expressed in terms of infinite sums or higher order derivatives of the fading parameter m.