Fourth-order Bessel-type special functions: a survey

TL;DR

This survey explores the properties of fourth-order Bessel-type functions, their differential equations, associated operators, and generalizations of the Hankel transform, building upon classical Bessel and Laguerre functions.

Contribution

It provides a comprehensive overview of the fourth-order Bessel-type functions, including their derivation, properties, and connections to classical special functions and transforms.

Findings

Characterization of the fourth-order Bessel-type differential equation

Construction of self-adjoint operators in Hilbert spaces

Generalization of the Hankel integral transform

Abstract

This survey paper reports on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform. These results are based upon the properties of the classical Bessel and Laguerre secondorder differential equations, and on the fourth-order Laguerre-type differential equation. From these differential equations and their solutions, limit processes yield the fourth-order Bessel-type functions and the associated differential equation.