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Lauricella hypergeometric function (LHF) FA is often used in the performance analysis of wireless communications, especially diversity systems. In this letter, FA is applied to solve two challenging problems: (1) a product of error functions averaged over generic fading channels and (2) error probability of BPSK modulation with equal-gain combining (EGC) in independent Nakagami-m fading channels. Closed-form expressions in terms of FA are obtained. In addition, a novel and efficient approximation to FA is presented.