The Ratio of the Generalized Marcum Functions of the First and Second Kind: Monotonicity and Bounds

Abstract

Let Qν(a,b) be the generalized Marcum Q-function and Rν(a,b) be the generalized Marcum function of the second kind of order ν>0. In this paper our aim is to present some new results related to the monotonicity and bounds on the ratio of the generalized Marcum functions of the first and second kinds Qν(a,b)/Rν(a,b). We also present new closed form series and integral representations for Rν(a,b) and its higher order derivatives. The results on the ratio of the generalized Marcum functions of the first and second kinds are based on some earlier established results on Qν(a,b) and Rν(a,b), and also on properties of modified Bessel functions of the first and second kind. Moreover, we show that the generalized Marcum function of the second kind is related to a distribution, which is not only infinitely divisible, but belongs also to the classes of self-decomposable distributions, generalized gamma convolutions and hyperbolically completely monotone densities. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2026.