  In this paper we focus on continuous univariate probability distributions,
like McKay distributions, $K$-distribution, generalized inverse Gaussian
distribution and generalised McKay distributions, with support $[0,\infty),$
which are related to modified Bessel functions of the first and second kinds
and in most cases we show that they belong to the class infinitely divisible
distributions, self-decomposable distributions, generalized gamma convolutions
and hyperbolically completely monotone densities. Some of the results are
known, however the proofs are new and we use special functions technique.
Integral representations of quotients of Tricomi hypergeometric functions as
well as of quotients of Gaussian hypergeometric functions, or modified Bessel
functions of the second kind play an important role in our study. In addition,
by using a different approach we rediscover a Stieltjes transform
representation due to Hermann Hankel for the product of modified Bessel
functions of the first and second kinds and we also deduce a series of new
Stieltjes transform representations for products, quotients and their
reciprocals concerning modified Bessel functions of the first and second kinds.
By using these results we obtain new infinitely divisible modified Bessel
distributions with Laplace transforms related to modified Bessel functions of
the first and second kind. Moreover, we show that the new Stieltjes transform
representations have some interesting applications and we list some open
problems which may be of interest for further research. In addition, we present
a new proof via the Pick function characterization theorem for the infinite
divisibility of the ratio of two gamma random variables and we present some new
Stieltjes transform representations of quotients of Tricomi hypergeometric
functions.
