Infinitely divisible modified Bessel distributions

Árpád Baricz, Dhivya Prabhu K, Sanjeev Singh and Antony Vijesh V

Vol. 343 (2026), No. 2, 261–313

DOI: 10.2140/pjm.2026.343.261

Abstract

We study certain continuous univariate probability distributions supported on $[0,ß)$ — the McKay distribution and its generalizations, the generalized inverse Gaussian distribution and the $K$-distribution — , all of which are related to modified Bessel functions of the first and second kinds. In most cases we show that they belong to the class of infinitely divisible distributions, self-decomposable distributions, generalized gamma convolutions and hyperbolically completely monotone densities. Some of the results are known, but new proofs are provided using special functions techniques: Integral representations of quotients of Tricomi hypergeometric functions, Gaussian hypergeometric functions, and modified Bessel functions of the second kind, play an important role in our study. In addition, by using a different approach based on asymptotic properties of modified Bessel functions, 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 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. We show that the new Stieltjes transform representations have some interesting applications and we list some open problems that may be of interest for further research. In addition, we present a new proof, using the Pick function characterization theorem, for the infinite divisibility of the ratio of two gamma random variables and some new Stieltjes transform representations of quotients of Tricomi hypergeometric functions.

Á. Baricz dedicates this paper to Mourad E. H. Ismail on the occasion of his 80th birthday

Keywords

modified Bessel functions, Tricomi hypergeometric function, infinitely divisible distribution, self-decomposable distribution, generalized gamma convolution, hyperbolically completely monotone distribution

Mathematical Subject Classification

Primary: 60E07, 44A15, 33C10

Milestones

Received: 8 October 2025

Revised: 2 February 2026

Accepted: 1 May 2026

Published: 22 May 2026