Identities and Properties of Multi-Dimensional Generalized Bessel Functions

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 Schlömilch series. We extend these results to mixed-type GBFs where appropriate.

Revised version of manuscript. Updated references. Paper will not be submitted to journal, but will remain here on arXiv as a reference

References in corpus (2)

• On two-dimensional Bessel functions
• Recursive algorithm for arrays of generalized Bessel functions: Numerical access to Dirac-Volkov solutions