Bounds for the generalized Marcum function of the second kind

Abstract

In this paper, we focus on the generalized Marcum function of the second kind of order ν> 0 , defined by Rν(a,b)=ca,νaν-1∫b∞tνe-t2+a22Kν-1(at)dt,where a> 0 , b≥ 0 , Kν stands for the modified Bessel function of the second kind, and ca,ν is a constant depending on a and ν such that Rν(a, 0) = 1. Our aim is to find some new tight bounds for the generalized Marcum function of the second kind and compare them with the existing bounds. In order to deduce these bounds, we include the monotonicity properties of various functions containing modified Bessel functions of the second kind as our main tools. Moreover, we demonstrate that our bounds in some sense are the best possible ones. © 2021, The Author(s).