A novel representation for the Nuttall Q-function

The aim of this work is the derivation of a novel explicit representation for the Nuttall Q-function, Qm,n(a, b). The Nuttall Q-function is a relatively new special function. Its usefulness in wireless communications is evident by its utilization in applications related to the field of digital communications over fading channels, [1]- and the references therein. It is denoted as Qm,n(a, b) and it was first proposed by A. H. Nuttall, in [2], as a generalisation of the Marcum Q-function, Qm(a, b), [3]–[5]. Consequently, M. K. Simon derived, in [6], a closed-form expression for Qm,n(a, b) in terms of Qm(a, b) and the modified Bessel function of the first kind, In(x). This representation has a finite series form and is valid only for the case that the sum of its two order indices -m + n- is an odd integer. In the same context, establishment of monotonicity criteria and derivation of tight bounds were given in [7] along with a useful closed-form expression which is valid for the case that m and n are odd multiples of 0.5, i.e m + 0.5 ∈ ℕ, n+ 0.5 ∈ ℕ.