A New Twist on the Generalized Marcum Q-Function QM(a, b) with Fractional-Order M and Its Applications

A new exponential-type integral for the generalized M-th order Marcum Q-function Q_M(alpha, beta) is obtained when M is not necessarily an integer. This new representation includes a classical formula due to Helstrom for the special case of positive integer order M and an additional integral correction term that vanishes when M assumes an integer value. The new form has both computational utility (numerous existing computational algorithms for Q_M(alpha, beta) are limited to integer M) and analytical utility (e.g., performance evaluation of selection diversity receiver in correlated Nakagami-m fading with arbitrary fading severity index, unified analysis of binary and quaternary modulations over generalized fading channels, and development of a Markovian threshold model for block errors in correlated Nakagami-m fading channels). Tight upper and lower bounds for Q_M(alpha, beta) that holds for any arbitrary real order M ges 0.5 are also derived.