A complex Lie algebra of rotationally symmetric operators and their harmonics
Markus Klintborg
TL;DR
The paper develops a unified framework for rotationally symmetric generalised harmonic functions in the complex plane, deriving canonical series representations from a four-dimensional complex Lie algebra. By reducing to one-dimensional ODEs, the authors connect the solutions to both confluent hypergeometric (Kummer) and Bessel-type (Θ) functions, and introduce a unifying object $oxed{ ext{𝒫}}$ that collapses to these classical forms depending on the parameter sum. They prove smoothness and absolute convergence of the series in the unit disk, establish asymptotic limits toward $e^{s z}$, and provide explicit coefficient formulas tied to derivatives at the origin. The results generalise harmonic expansions beyond Laplace/Helmholtz, offering a structured, rotation-based decomposition with clear convergence and asymptotics, relevant for generalized harmonic analysis in symmetric operator algebras.
Abstract
We describe the solutions to a family of rotationally symmetric second order partial differential equations in the complex plane that arises from a four-dimensional complex Lie algebra whose spanning set generates the algebra from which such generalised harmonic functions derive. We show that every one of these solutions have a canonical series representation and retrieve those obtained in the case of Laplace and Helmholtz equation. These sums are given in confluent hypergeometric terms that asymptotically correspond to the complex exponential function.