Motivated by recent results by Nadarajah, Srivastava and Gupta ( 2007 Nadarajah, S. , H. M. Srivastava, and A. K. Gupta. 2007. Skewed Bessel function distributions with application to rainfall data. Statistics 41 (4):333–44. doi: 10.1080/02331880701270606. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] ) in which the authors introduced a new formula for CDF F_K of the McKay Bessel function distribution containing the modified Bessel function $K_{\nu }$ we aim to present new formulae for F_K, especially for certain values of parameters for which their formula fails to be calculated by in-built numerical routines. Also, several computational series formulae are obtained for the CDF of McKay Bessel distribution which contains modified Bessel function $I_{\nu };$ one is given in terms of the Horn confluent hypergeometric function ${\Phi }_2,$ and the others by Kampé de Fériet function $F_{l:m;n}^{p:q;k};$ Marcum Q-function and modified Bessel functions $I_{\nu };$ Horn confluent hypergeometric function ${\Phi }_3;$ and finally in terms of incomplete Gamma function γ and Exton’s double hypergeometric X–function.

Recently Pogány and Süli [Proc. Amer. Math. Soc. 137(7) (2009), 2363-2368] derived a closed-form integral expression for Neumann series of Bessel functions of the first kind . In this paper our aim is to establish analogous integral representations for the Neumann-type series of modified Bessel functions of the first kind and for Bessel functions of the second kind <!-- MATH $Y_\nu , K_\nu$ --> , and to give links for the same question for the Hankel functions <!-- MATH $H_\nu ^{(1)}, H_\nu ^{(2)}$ --> .

The main aim of this research note is to establish two different types of integral representation formulae for the Kapteyn series of the first kind. The first one is a double definite integral expression, while the second type includes indefinite integral representation formula.