Identities and Properties of Multi-Dimensional Generalized Bessel Functions

TL;DR

This paper explores the properties of multi-dimensional generalized Bessel functions, including their differential equations, asymptotic behaviors, and series expansions, extending known results to higher dimensions and mixed types.

Contribution

It provides new insights into the structure and properties of m-dimensional GBFs, including PDEs, asymptotics, and series analysis, expanding the theoretical understanding of these functions.

Findings

Derived partial differential equation structure for m-dimensional GBFs

Established asymptotic behaviors for large order and argument

Analyzed generalized Neumann, Kapteyn, and Schlömilch series for GBFs

Abstract

The Generalized Bessel Function (GBF) extends the single variable Bessel function to several dimensions and indices in a nontrivial manner. Two-dimensional GBFs have been studied extensively in the literature and have found application in laser physics, crystallography, and electromagnetics. In this article, we document several properties of $m$-dimensional GBFs including an underlying partial differential equation structure, asymptotics for simultaneously large order and argument, and analysis of generalized Neumann, Kapteyn, and Schl\"{o}milch series. We extend these results to mixed-type GBFs where appropriate.