The McKay $$I_\nu $$ Bessel distribution revisited

Abstract

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.