Novel expressions for the Marcum and one dimensional Q-functions

The aim of this work is twofold. Firstly, we derive a novel closed-form representation for the Marcum Q-function, Qm(a, b), which is valid for all values of its order m. Secondly, we propose a novel tight approximation for the one dimensional Q-function, Q(x). The high accuracy of the derived expressions is verified with the aid of computational algorithms as well as through comparisons with existing, yet limited, representations. Importantly, the derived relationships have a relatively simple algebraic form and therefore, they are convenient to handle both analytically and numerically. As a consequence, they may be considered as useful mathematical tools in analytic performance evaluations of digital communications over fading channels.