On the Monotonicity of the Generalized Marcum and Nuttall ${Q}$ -Functions

Monotonicity criteria are established for the generalized Marcum
Q
-function,
QM
(alpha,beta), the standard Nuttall
Q
-function,
QM
,
N
(alpha,beta) , and the normalized Nuttall
Q
-function,
QM
,
N
(alpha,beta), with respect to their real order indices
M
,
N
. Besides, closed-form expressions are derived for the computation of the standard and normalized Nuttall
Q
-functions for the case when
M
,
N
are odd multiples of 0.5 and
M
ges
N
. By exploiting these results, novel upper and lower bounds for
QM
,
N
(alpha,beta) and
QM
,
N
(alpha,beta) are proposed. Furthermore, specific tight upper and lower bounds for
QM
(alpha,beta), previously reported in the literature, are extended for real values of
M
. The offered theoretical results can be efficiently applied in the study of digital communications over fading channels, in the information-theoretic analysis of multiple-input multiple-output systems and in the description of stochastic processes in probability theory, among others.