The Humbert-Bessel functions, Stirling numbers and probability distributions in coincidence problems

TL;DR

This paper explores the relationships between Humbert-Bessel functions, re-modified Bessel functions, and Stirling numbers, demonstrating how these special functions relate to probability distributions in coincidence problems.

Contribution

It establishes that re-modified Bessel functions are specific cases of Humbert-Bessel functions, clarifying their properties within probability theory.

Findings

Re-modified Bessel functions are expressible in terms of Humbert-Bessel functions.

Humbert-Bessel functions relate to probability distributions in coincidence problems.

The paper unifies various special functions under the Humbert-Bessel framework.

Abstract

The Humbert-Bessel are multi-index functions with various applications in electromagnetism. New families of functions sharing some similarities with Bessel functions are often introduced in the mathematical literature, but at a closer analysis they are not new, in the strict sense of the word, and are shown to be expressible in terms of already discussed forms. This is indeed the case of the re-modified Bessel functions, whose properties have been analyzed within the context of coincidence problems in probability theory. In this paper we show that these functions are particular cases of the Humbert-Bessel ones.