Bearing in mind an increasing popularity of the fractional calculus the main aim of this paper is to derive several new representation formulae for the cumulative distribution function (cdf) of the McKay \(I_\nu \) Bessel distribution including the Grünwald-Letnikov fractional derivative; also, two connection formulae between cdf of the McKay \(I_\nu \) random variable and the so–called Neumann series of modified Bessel functions of the first kind are established, providing, consequently, a new integral representation for such cdf in terms of a definite integral. Another fashion expression for the given cdf is derived in terms of the Grünwald-Letnikov fractional derivative of the widely applicable Marcum Q–function, which represents a certain simplification of the already existing relationship between McKay \(I_\nu \) random variable and a Marcum Q–functions. The exposition ends with some open questions, drawing the interested reader’s attention, among others, to the summation of some Neumann series.

考虑到分数阶微积分的日益普及，本文的主要目的是推导包括 Grünwald-Letnikov 分数导数的 McKay \(I_\nu \) Bessel 分布的累积分布函数 (cdf) 的几个新表示公式;此外，还建立了 McKay \(I_\nu \)随机变量的 cdf 与所谓的第一类修正 Bessel 函数的 Neumann 级数之间的两个联系公式，从而为此类 cdf 提供了一种新的积分表示形式的定积分。给定 cdf 的另一种时尚表达式是根据广泛适用的 Marcum Q 函数的 Grünwald-Letnikov 分数导数导出的，它表示 McKay \(I_\nu \)随机变量和 a 之间已经存在的关系的某种简化。 Marcum Q——函数。论述以一些开放性问题结束，吸引感兴趣的读者的注意力，其中包括一些诺伊曼级数的总结。