Constructing steering-type solutions for higher order Cauchy-Riemann equations in $\mathbb{R}^{m+1}$

TL;DR

This work extends steering techniques from complex CR theory to higher-order Cauchy–Riemann systems within Clifford analysis, constructing two-sided, polymonogenic and polyharmonic solutions by coupling complex-valued steering families with Clifford-valued factors. Across exponential, trigonometric, and power steering, the authors derive explicit monogenicity conditions, revealing harmonic, polyharmonic, and derivative-structured relationships that govern left, right, and two-sided monogenicity. They show how to generate n-monogenic solutions via Δ^n A = 0 with related B-corrections, and connect these constructions to hypercomplex derivatives to solve homogeneous differential equations, including Appell-type and inframonogenic generalizations. The results yield a versatile toolkit for building steering-type solutions to a broad class of higher-dimensional CR-type systems, with links to physical models like Lamé-Navier and potential universal solutions. Overall, the paper provides explicit, structured methods to generate and analyze higher-order monogenic and polymonogenic functions in ℝ^{m+1}.

Abstract

The multidimensional Cauchy-Riemann operator provides a framework for studying higher order partial differential equations in $\mathbb{R}^{m+1}$, whose solutions include polymonogenic and polyharmonic functions, among others. In this work, we aim to explicitly construct solutions to such systems, generated from families of complex valued functions which are closed under conjugation and under the action of the complex Cauchy-Riemann operator. Moreover, we prove that precisely some of these solutions also satisfy homogeneous linear differential equations involving the so-called hypercomplex derivative.

Theorems & Definitions (13)

• Definition 1
• Remark 1
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 2
• Theorem 4
• Remark 3
• Proposition 1
• Remark 4