The Ramanujan Journal ( IF 0.6 ) Pub Date : 2021-06-05 , DOI: 10.1007/s11139-021-00440-9 Árpád Baricz , Nitin Bisht , Sanjeev Singh , V. Antony Vijesh
In this paper, we focus on the generalized Marcum function of the second kind of order \(\nu >0\), defined by
$$\begin{aligned} R_{\nu }(a,b)=\frac{c_{a,\nu }}{a^{\nu -1}} \int _b ^ {\infty } t^{\nu } e^{-\frac{t^2+a^2}{2}}K_{\nu -1}(at)\mathrm{d}t, \end{aligned}$$where \(a>0, b\ge 0,\) \(K_{\nu }\) stands for the modified Bessel function of the second kind, and \(c_{a,\nu }\) is a constant depending on a and \(\nu \) such that \(R_{\nu }(a,0)=1.\) 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.
中文翻译：
第二类广义 Marcum 函数的界
在本文中，我们关注二阶\(\nu >0\)的广义 Marcum 函数，定义为
$$\begin{aligned} R_{\nu }(a,b)=\frac{c_{a,\nu }}{a^{\nu -1}} \int _b ^ {\infty } t^{ \nu } e^{-\frac{t^2+a^2}{2}}K_{\nu -1}(at)\mathrm{d}t, \end{aligned}$$其中\(a>0, b\ge 0,\) \(K_{\nu }\)代表第二类修正贝塞尔函数，\(c_{a,\nu }\)是一个常数，取决于在a和\(\nu \)使得\(R_{\nu }(a,0)=1.\)我们的目标是为第二类广义 Marcum 函数找到一些新的紧边界，并将它们与现有的界限。为了推导出这些界限，我们将包含第二类修正贝塞尔函数的各种函数的单调性作为我们的主要工具。此外，我们证明了我们的边界在某种意义上是最好的。
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