On Corresponding Cauchy–Riemann Equations Applied to Laplace-Type Operators over Generalized Quaternions, with an Application

Abstract

In this paper, we develop a concise differential–potential framework for the functions of a generalized quaternionic variable in the two-parameter algebra ${\mathbb{H}}_{\alpha ,\beta }$, with $\alpha ,\beta \in \mathbb{R}\setminus \left\{0\right\}$. Starting from left/right difference quotients, we derive complete Cauchy–Riemann (CR) systems and prove that, away from the null cone where the reduced norm N vanishes, these first-order systems are necessary and, under ${\mathcal{C}}^1$ regularity, sufficient for left/right differentiability, thereby linking classical one-dimensional calculus to a genuinely four-dimensional setting. On the potential theoretic side, the Dirac factorization ${\Delta }_{\alpha ,\beta }=\overline{\mathcal{D}}\mathcal{D}=\mathcal{D}\overline{\mathcal{D}}$ shows that each real component of a differentiable mapping is ${\Delta }_{\alpha ,\beta }$-harmonic, yielding a clean second-order theory that separates the elliptic (Hamiltonian) and split (coquaternionic) regimes via the principal symbol. In the classical case $(\alpha ,\beta )=(-1,-1)$, we present a Poisson-type representation solving a model Dirichlet problem on the unit ball $B\subset {\mathbb{R}}^4$, recovering mean-value and maximum principles. For computation and symbolic verification, real $4\times 4$ matrix models for left/right multiplication linearize the CR systems. Examples (polynomials, affine CR families, and split-signature contrasts) illustrate the theory, and the outlook highlights boundary integral formulations, Green kernel constructions, and discretization strategies for quaternionic PDEs.

1. Introduction

The Laplace equation is a cornerstone of potential theory and a unifying language for equilibrium phenomena in physics and engineering. In the planar case, a scalar potential u satisfies

$${\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}=0,$$

(1)

and in higher dimensions, $\Delta u=0$. Its solutions harmonic functions are smooth and obey maximum and mean-value principles, explaining their pervasive role in gravitation, electrostatics, and incompressible, irrotational fluid flow.

In non-commutative settings such as quaternions and their generalizations, first-order systems that play the role of the complex Cauchy–Riemann (CR) equations typically force second-order Laplace-type relations for the real components. This article develops that paradigm in the generalized quaternion algebra ${\mathbb{H}}_{\alpha ,\beta }$ with non-zero parameters $\alpha ,\beta \in \mathbb{R}$, where

$$i^2=\alpha ,j^2=\beta ,ij=k=-ji,k^2=-\alpha \beta .$$

(2)

We study mappings $f:{\mathbb{H}}_{\alpha ,\beta }\to {\mathbb{H}}_{\alpha ,\beta }$ via left/right difference quotients and show how first-order quaternionic analyticity interfaces with second-order potential theory. Our approach unifies algebraic, analytic, and matrix-linear viewpoints and is designed to be verifiable by direct symbolic computation.

• Scope and goals (compact overview)

We study mappings $f:{\mathbb{H}}_{\alpha ,\beta }\to {\mathbb{H}}_{\alpha ,\beta }$ for the algebra defined in (2). Our goals are minimal yet complete: (i) to obtain left/right CR systems from differentiability; (ii) to prove sufficiency off the null cone; (iii) connect first-order CR structure to second-order potential theory via a Dirac factorization; (iv) tp give a clean Poisson representation in the elliptic case; and (v) to package the calculus in $4\times 4$ real matrices for symbolic and numerical verification.

• Setting and viewpoint.

Writing $f=u_0+i{u}_1+j{u}_2+k{u}_3$ with real components $u_r$, we derive CR-type identities from the existence of left/right derivatives and prove that each $u_r$ solves a Laplace-type equation governed by $(\alpha ,\beta )$. The analysis is carried out on open sets $U\subset {\mathbb{R}}^4$ that avoid the vanishing locus of the reduced norm (the null cone $\mathcal{K}$) with respect to the topology induced by the modulus associated with ${\mathbb{H}}_{\alpha ,\beta }$. This links first-order structure to second-order harmonicity in a manner that generalizes the classical complex theory.

• Contributions

We summarize the main results in a concise, computation-ready form:

• CR systems and differentiability: Starting from left/right difference quotients, we derive complete CR systems and prove necessity and—on $U\setminus \mathcal{K}$ under ${\mathcal{C}}^1$ regularity—sufficiency for left/right differentiability. This creates a precise bridge between one-dimensional calculus and a genuinely four-dimensional setting.
• Dirac factorization and harmonicity: We factor a constant coefficient second-order operator

  $${\Delta }_{\alpha ,\beta }={\partial }_{x_0}^2-\alpha {\partial }_{x_1}^2-\beta {\partial }_{x_2}^2+\alpha \beta {\partial }_{x_3}^2=\overline{\mathcal{D}}\mathcal{D}=\mathcal{D}\overline{\mathcal{D}},$$

  (3)

  which implies that each real component of a differentiable mapping is ${\Delta }_{\alpha ,\beta }$-harmonic. The principal symbol cleanly separates the elliptic (Hamiltonian) and split (coquaternionic) regimes.
• Model boundary value problem: In the elliptic case $(\alpha ,\beta )=(-1,-1)$, we establish a Poisson-type representation that solves a model Dirichlet problem on the unit ball $B\subset {\mathbb{R}}^4$, recovering the mean value and maximum principles.
• Linearization and computation: Real $4\times 4$ matrix models for left/right multiplication linearize the CR systems and the associated second-order operator, yielding identities suitable for symbolic or numerical verification.
• Examples and contrasts: A portfolio of examples (polynomials, affine CR families, and split-signature contrasts) illustrates the theory and highlights the role of signature.

• Related background

Our formulation is informed by several strands of work. On product domains, Chakrabarti and Shaw [1] developed an $L^2$ theory for the CR operator with a closed range on each factor. Early hypercomplex approaches to PDE go back to Cockle and, later, to Clifford analysis, where generalized CR and iterated Dirac systems were studied [2,3,4]. Bicomplex holomorphy has been systematized in [5]. Kajiwara and collaborators related complex-valued harmonic functions on pseudoconvex domains to quaternionic hyperholomorphy under integrability assumptions [6]. Krantz highlighted deep links between PDE and the function theory of several complex variables [7]. Within quaternionic structures, Kim developed algebraic/analytic aspects of Pauli quaternions, including polar form and de Moivre’s formula via Pauli matrix isomorphisms [8]; Kim-Shon analyzed differentiability in generalized quaternionic fields via Riemann-type conditions [9]. Marão–Neto derived Laplace-like equations for quaternion variables from Riemann–Cauchy hypercomplex relations [10]. The HR calculus provides a practical derivative framework based on involutions and real-vector isomorphisms [11]. At a geometric level, Haydys connected the harmonic spinors of generalized non-linear Dirac operators with Cauchy–Riemann–Fueter equations on quaternionic manifolds [12]. Sudbery’s classification of regular quaternion-valued functions clarified the analogies with the complex theory [13].

• Recent literature (2022–2025)

To situate the present work within current activity, we briefly highlight several complementary strands. On the analytic–physical interface, split quaternionic formulations of Maxwell/Dirac–type systems sharpen the link between hypercomplex analyticity and first-order PDE models [14]. On the computational side, structure-preserving algorithms for split-quaternionic Schrödinger equations broaden the numerical toolkit and point to stable, higher-order evolution solvers [15,16]. In integral representation theory, fractional and generalized Borel–Pompeiu formulas, together with the $\psi$-Fueter operator, extend the classical kernel calculus [17], while the $(q,q^{\prime })$-Borel–Pompeiu model further systematizes such deformations and parameterized identities [18]. At a foundational level, generalized Cauchy–Riemann systems expressed in non-identity bases clarify basis dependence and invariance issues in hypercomplex analysis [19]. Discrete potential theoretic methods for conjugate harmonics enrich the hypercomplex toolbox with graph/discrete counterparts [20]. Operator theoretic advances on quaternion integral transforms illuminate spectral and mapping properties relevant to PDE and signal analysis [21]. Finally, the burgeoning several variables theory for slice-regular functions, including results on zero sets and algebraic structure, signals a widening scope toward genuinely multivariate phenomena [22].

Positioning of the present paper: These developments collectively underscore (i) the utility of first-order Dirac/CR frameworks for governing equations; (ii) the reach of generalized Borel–Pompeiu calculi for representation and boundary methods; and (iii) the importance of basis-aware formulations and discrete/algorithmic perspectives. The framework advanced here over the two–parameter algebra ${\mathbb{H}}_{\alpha ,\beta }$ complements these directions by deriving complete left/right CR systems with a Dirac factorization valid across elliptic and split regimes, furnishing boundary-integral tools aligned with generalized Borel–Pompeiu ideas and clarifying the role of basis changes via explicit matrix models and principal symbol analysis.

• Organization

Section 2 fixes the algebraic and analytic background in the two-parameter algebra ${\mathbb{H}}_{\alpha ,\beta }$. We record the bilinear product with the corrected j-component (5), introduce conjugation, reduced norm and modulus, and identify the null cone $\mathcal{K}$ and its signature via the quadratic form (10) (see Section 2.3). We then present real $4\times 4$ matrix models for left/right multiplication (11), together with determinant, trace, and pseudo-orthogonality identities (12) and (13) (Section 2.4), fix the component notation for mappings (14) (Section 2.5), and record the Dirac/Fueter operators (15), which take into account the constant coefficient second-order operator (16) (Section 2.6). Section 3 contains the core results: in Section 3.1, we derive the left/right CR systems (20) and (21) and prove their necessity and (under ${\mathcal{C}}^1$ regularity off $\mathcal{K}$) sufficiency (Theorem 1). In Section 3.2, we deduce the ${\Delta }_{\alpha ,\beta }$ harmonicity of the components (Theorem 2) and characterize elliptic versus split regimes via the principal symbol (Proposition 1). The elliptic case $(\alpha ,\beta )=(-1,-1)$ is illustrated in Section 3.3 via a model Dirichlet problem and Poisson representation (Theorem 4). Examples and consistency checks appear in Section 3.4. Section 4 outlines the (Borel–Pompeiu-type) boundary integral formulations, Green kernels in the split signature, weak formulations on Sobolev scales, and structure-preserving numerics, exploiting matrix linearizations. Section 5 closes with the scope and limitations of this study near the null cone and looks forward to boundary integral formulations, Green kernels, weak formulations, and structure-preserving discretizations.

2. Preliminaries and Notation

2.1. The Generalized Quaternion Algebra ${\mathbb{H}}_{\alpha ,\beta }$

Fix real parameters $\alpha ,\beta \in \mathbb{R}\setminus \left\{0\right\}$. We work in real, associative, four dimensional algebra

$${\mathbb{H}}_{\alpha ,\beta }={span}_{\mathbb{R}}\{1,i,j,k\},i^2=\alpha ,j^2=\beta ,ij=k=-ji,k^2=-\alpha \beta ,$$

(4)

which, in particular, implies

$$jk=-\beta i=-kj,ki=-\alpha j=-ik.$$

Every element $p\in {\mathbb{H}}_{\alpha ,\beta }$ is uniquely written as

$$p=x_0+i{x}_1+j{x}_2+k{x}_3,x_r\in \mathbb{R}(r=0,1,2,3).$$

Real vector space structure

Throughout, we regard ${\mathbb{H}}_{\alpha ,\beta }$ as a real algebra: addition is component-wise in ${\mathbb{R}}^4$, and real scalar multiplication acts diagonally. Unless stated otherwise, all topological and differentiability notions are presented with respect to this real structure.

Remark 1

(Complexification). We outline ${\mathbb{H}}_{\alpha ,\beta }{\otimes }_{\mathbb{R}}\mathbb{C}$. Note that $\mathcal{D}$ and $\overline{\mathcal{D}}$ extend $\mathbb{C}$-linearly, but the left/right differentiability notion changes; precise analysis is deferred to future work and briefly contrasted with the real case.

2.2. Bilinear Expansion of the Product

For $p=x_0+i{x}_1+j{x}_2+k{x}_3$ and $q=y_0+i{y}_1+j{y}_2+k{y}_3$, multiplication is bilinear and generally non-commutative:

$$\begin{array}{cc}\hfill pq& =(x_0{y}_0+\alpha x_1{y}_1+\beta x_2{y}_2-\alpha \beta x_3{y}_3)\hfill \\ \hfill & +i(x_0{y}_1+x_1{y}_0-\beta x_2{y}_3+\beta x_3{y}_2)\hfill \\ \hfill & +j(x_0{y}_2+x_2{y}_0+\alpha x_1{y}_3-\alpha x_3{y}_1)\hfill \\ \hfill & +k(x_0{y}_3+x_3{y}_0+x_1{y}_2-x_2{y}_1).\hfill \end{array}$$

(5)

2.3. Conjugation, Reduced Norm, Modulus, Inverse, and the Null Cone

Define the (anti-involutive) conjugation and the reduced norm by

$$p^{*}:=x_0-i{x}_1-j{x}_2-k{x}_3,N\left(p\right):=p{p}^{*}=p^{*}p=x_0^2-\alpha x_1^2-\beta x_2^2+\alpha \beta x_3^2.$$

(6)

Then, ${\left(pq\right)}^{*}=q^{*}{p}^{*}$ and $N\left(pq\right)=N\left(p\right)N\left(q\right)$. We also use the (non-negative) modulus

$$M\left(p\right):=\sqrt{\left|N\right(p\left)\right|},|p{p}^{*}|=|p^{*}{p|=M\left(p\right)}^2.$$

(7)

The set of non–invertible elements is the quadratic cone:

$$\mathcal{K}:=\{p\in {\mathbb{H}}_{\alpha ,\beta }:N\left(p\right)=0\}=\left\{x_0^2-\alpha x_1^2-\beta x_2^2+\alpha \beta x_3^2=0\right\}.$$

(8)

For $p\notin \mathcal{K}$, the inverse exists and is given by

$$p^{-1}={\displaystyle \frac{p^{*}}{N\left(p\right)}}={\displaystyle \frac{x_0-i{x}_1-j{x}_2-k{x}_3}{x_0^2-\alpha x_1^2-\beta x_2^2+\alpha \beta x_3^2}}.$$

(9)

Quadratic form and signature

Write $x={(x_0,x_1,x_2,x_3)}^{\top }$. Then, $N\left(p\right)=x^{\top }{G}_{\alpha ,\beta }x$, with

$$G_{\alpha ,\beta }:=\mathrm{diag}\left(1,-\alpha ,-\beta ,\alpha \beta \right).$$

(10)

In particular, when $\alpha =\beta =-1$, we have $G_{\alpha ,\beta }=I_4$. More generally, for $\alpha <0$ and $\beta <0$, all diagonal entries of $G_{\alpha ,\beta }=\mathrm{diag}(1,-\alpha ,-\beta ,\alpha \beta )$ are positive, so $G_{\alpha ,\beta }$ is a positive definite (indeed, it is congruent to $I_4$ via diagonal rescaling).

• Topology

We identify ${\mathbb{H}}_{\alpha ,\beta }\cong {\mathbb{R}}^4$ with its Euclidean topology. The function M is continuous and, on compact subsets of ${\mathbb{H}}_{\alpha ,\beta }\setminus \mathcal{K}$, equivalent to the Euclidean norm; hence, both induce the same local notion of limits and differentiability away from $\mathcal{K}$.

2.4. Left/Right Multiplication as Real Matrices

We denote left and right multiplication by $L_p\left(q\right):=pq$ and $R_p\left(q\right):=qp$. With respect to the basis $(1,i,j,k)$ m, these $\mathbb{R}$-linear maps are represented by

$$L_p=\left(\begin{array}{cccc}{x}_0& \alpha x_1& \beta x_2& -\alpha \beta x_3\\ x_1& x_0& \beta x_3& -\beta x_2\\ x_2& -\alpha x_3& x_0& \alpha x_1\\ x_3& -x_2& x_1& x_0\end{array}\right),R_p=\left(\begin{array}{cccc}{x}_0& x_1& x_2& x_3\\ \alpha x_1& x_0& \alpha x_3& x_2\\ \beta x_2& -\beta x_3& x_0& -x_1\\ -\alpha \beta x_3& \beta x_2& -\alpha x_1& x_0\end{array}\right).$$

(11)

Associativity gives $L_p{R}_q=R_q{L}_p$ for all $p,q\in {\mathbb{H}}_{\alpha ,\beta }$. Multiplicativity of the norm yields

$$det{L}_p=det{R}_p={\left(N\left(p\right)\right)}^2,\mathrm{tr}{L}_p=\mathrm{tr}{R}_p=4x_0.$$

(12)

A useful pseudo0orthogonality identity follows from $N\left(pq\right)=N\left(p\right)N\left(q\right)$:

$$L_p^{\top }{G}_{\alpha ,\beta }{L}_p=N\left(p\right)G_{\alpha ,\beta },R_p^{\top }{G}_{\alpha ,\beta }{R}_p=N\left(p\right)G_{\alpha ,\beta }.$$

(13)

In particular, when $\alpha =\beta =-1$ (so $G_{\alpha ,\beta }=I_4$), one has $L_p^{\top }{L}_p=R_p^{\top }{R}_p=N\left(p\right)I_4$.

2.5. Notation for Functions and Differentials

Let $U\subset {\mathbb{H}}_{\alpha ,\beta }$ be open. A function $f:U\to {\mathbb{H}}_{\alpha ,\beta }$ is written in components as

$$f\left(x\right)=u_0\left(x\right)+i{u}_1\left(x\right)+j{u}_2\left(x\right)+k{u}_3\left(x\right),\mathbf{u}\left(x\right):={(u_0,u_1,u_2,u_3)}^{\top },$$

(14)

with $u_r:U\to \mathbb{R}$. We write $u_{r,t}:={\partial }_{x_t}{u}_r$ for the first partials and $Jf$ for the real $4\times 4$ Jacobian of f. We also use $\Re \left(p\right):=x_0$ and $\Im \left(p\right):=i{x}_1+j{x}_2+k{x}_3$.

2.6. Dirac/Fueter Operators and a Constant Coefficient Factorization

We record the first-order operators

$$\mathcal{D}:={\partial }_{x_0}-i{\partial }_{x_1}-j{\partial }_{x_2}-k{\partial }_{x_3},\overline{\mathcal{D}}:={\partial }_{x_0}+i{\partial }_{x_1}+j{\partial }_{x_2}+k{\partial }_{x_3},$$

(15)

which, by (4) and (5), commute on scalar fields and factor the constant coefficient second-order operator

$${\Delta }_{\alpha ,\beta }:=\overline{\mathcal{D}}\mathcal{D}=\mathcal{D}\overline{\mathcal{D}}={\partial }_{x_0}^2-\alpha {\partial }_{x_1}^2-\beta {\partial }_{x_2}^2+\alpha \beta {\partial }_{x_3}^2.$$

(16)

The principal symbol of ${\Delta }_{\alpha ,\beta }$ is ${\xi }^{\top }{G}_{\alpha ,\beta }\xi$, matching the quadratic form in (10).

Remark 2

(Alternative symbols and consistency). We emphasize that boldface glyphs $(\mathbf{i},\mathbf{j},\mathbf{k})$, when used, denote the same units obeying (4). No new algebraic objects are introduced.

As a geometric guide, Figure 1 illustrates the null cone in the slice $x_3=0$ for $(\alpha ,\beta )=(-1,1)$.

3. Main Results

We work with the generalized quaternion algebra ${\mathbb{H}}_{\alpha ,\beta }$ of Section 2 with fixed parameters $\alpha ,\beta \in \mathbb{R}\setminus \left\{0\right\}$. Points are written as

$$p=x_0+i{x}_1+j{x}_2+k{x}_3,x=(x_0,x_1,x_2,x_3)\in {\mathbb{R}}^4,$$

and limits are taken in the M topology induced by the reduced norm M (equivalently, the Euclidean topology away from the null cone $\mathcal{K}=\{M=0\}$). Let $U\subset {\mathbb{H}}_{\alpha ,\beta }$ be open and $f\in {\mathcal{C}}^1(U,{\mathbb{H}}_{\alpha ,\beta })$ with component vector

$$f\left(x\right)=u_0\left(x\right)+i{u}_1\left(x\right)+j{u}_2\left(x\right)+k{u}_3\left(x\right),\mathbf{u}:={(u_0,u_1,u_2,u_3)}^{\top }.$$

(17)

3.1. Left/Right Differentiability and CR–Type Systems

Definition 1

(Left/right derivative). For $p_0\in U$, the left and rightderivatives of f at $p_0$ are the limits

$$f_L^{\prime }\left(p_0\right):=\underset{p\to p_0}{lim}{(p-p_0)}^{-1}\left(f\left(p\right)-f\left(p_0\right)\right),f_R^{\prime }\left(p_0\right):=\underset{p\to p_0}{lim}\left(f\left(p\right)-f\left(p_0\right)\right){(p-p_0)}^{-1},$$

(18)

whenever they exist in ${\mathbb{H}}_{\alpha ,\beta }$. We say f is left/right differentiable on U if the corresponding limit exists at every $p_0\in U$.

Using the multiplication rules (4), the existence of $f_L^{\prime }\left(p_0\right)$ is equivalent to the directional identity

$${\partial }_{x_0}f={\displaystyle \frac{i}{\alpha }}{\partial }_{x_1}f={\displaystyle \frac{j}{\beta }}{\partial }_{x_2}f=-{\displaystyle \frac{k}{\alpha \beta }}{\partial }_{x_3}f\mathrm{at}{p}_0,$$

(19)

and similarly for the right derivative with the factors on the right. Equating scalar and $i,j,k$ parts yields the first-order Cauchy–Riemann (CR)-type systems below.

Theorem 1

(Left/right CR systems: necessity and sufficiency). Let $U\subset {\mathbb{H}}_{\alpha ,\beta }$ be open and $f\in {\mathcal{C}}^1(U,{\mathbb{H}}_{\alpha ,\beta })$.

1. 

(Necessity) If f has a left (resp. right) derivative at $p_0\in U$, then the following left (resp. right) CR system holds at $p_0$:

$$\begin{array}{cc}\hfill & {\partial }_{x_0}{u}_0={\partial }_{x_1}{u}_1={\partial }_{x_2}{u}_2={\partial }_{x_3}{u}_3,\hfill \end{array}$$

(20a)

$$\begin{array}{cc}\hfill & {\partial }_{x_0}{u}_1=\frac{1}{\alpha }{\partial }_{x_1}{u}_0=-{\partial }_{x_2}{u}_3=-\frac{1}{\alpha }{\partial }_{x_3}{u}_2,\hfill \end{array}$$

(20b)

$$\begin{array}{cc}\hfill & {\partial }_{x_0}{u}_2={\partial }_{x_1}{u}_3=\frac{1}{\beta }{\partial }_{x_2}{u}_0=\frac{1}{\beta }{\partial }_{x_3}{u}_1,\hfill \end{array}$$

(20c)

$$\begin{array}{cc}\hfill & {\partial }_{x_0}{u}_3=\frac{1}{\alpha }{\partial }_{x_1}{u}_2=-\frac{1}{\beta }{\partial }_{x_2}{u}_1=-\frac{1}{\alpha \beta }{\partial }_{x_3}{u}_0.\hfill \end{array}$$

(20d)

$$\begin{array}{cc}\hfill & {\partial }_{x_0}{u}_0={\partial }_{x_1}{u}_1={\partial }_{x_2}{u}_2={\partial }_{x_3}{u}_3,\hfill \end{array}$$

(21a)

$$\begin{array}{cc}\hfill & {\partial }_{x_0}{u}_1=\frac{1}{\alpha }{\partial }_{x_1}{u}_0={\partial }_{x_2}{u}_3=\frac{1}{\alpha }{\partial }_{x_3}{u}_2,\hfill \end{array}$$

(21b)

$$\begin{array}{cc}\hfill & {\partial }_{x_0}{u}_2=-{\partial }_{x_1}{u}_3=\frac{1}{\beta }{\partial }_{x_2}{u}_0=-\frac{1}{\beta }{\partial }_{x_3}{u}_1,\hfill \end{array}$$

(21c)

$$\begin{array}{cc}\hfill & {\partial }_{x_0}{u}_3=-\frac{1}{\alpha }{\partial }_{x_1}{u}_2=\frac{1}{\beta }{\partial }_{x_2}{u}_1=-\frac{1}{\alpha \beta }{\partial }_{x_3}{u}_0.\hfill \end{array}$$

(21d)

2. 

(Sufficiency) Conversely, if the left (resp. right) system (20) (resp. (21)) holds on a neighborhood of $p_0\in U\setminus \mathcal{K}$, then f is left (resp. right) differentiable at $p_0$, and

$$f_L^{\prime }\left(p_0\right)={\partial }_{x_0}f\left(p_0\right)={\displaystyle \frac{i}{\alpha }}{\partial }_{x_1}f\left(p_0\right)={\displaystyle \frac{j}{\beta }}{\partial }_{x_2}f\left(p_0\right)=-{\displaystyle \frac{k}{\alpha \beta }}{\partial }_{x_3}f\left(p_0\right),$$

$$f_R^{\prime }\left(p_0\right)={\partial }_{x_0}f\left(p_0\right)={\partial }_{x_1}f\left(p_0\right){\displaystyle \frac{i}{\alpha }}={\partial }_{x_2}f\left(p_0\right){\displaystyle \frac{j}{\beta }}=-{\partial }_{x_3}f\left(p_0\right){\displaystyle \frac{k}{\alpha \beta }}.$$

Corollary 1

(CR sufficiency; concise form). If $f\in {\mathcal{C}}^1$ satisfies the left (resp. right) CR system on a neighborhood of $p_0\in U\setminus \mathcal{K}$, then f is left (resp. right) differentiable at $p_0$ and $f_L^{\prime }\left(p_0\right)={\partial }_{x_0}f\left(p_0\right)$ (resp. $f_R^{\prime }\left(p_0\right)={\partial }_{x_0}f\left(p_0\right)$).

Remark 3

(Explicit generators). We use $L_i=L_{(0,1,0,0)}$, $L_j=L_{(0,0,1,0)}$, and $L_k=L_{(0,0,0,1)}$ and, analogously, $R_i,R_j,and{R}_k$ obtained from (11). Then, (20) and (21) are equivalent to the compact relations

$$\begin{array}{c}\hfill {\partial }_{x_0}u={\displaystyle \frac{1}{\alpha }}{L}_i{\partial }_{x_1}u={\displaystyle \frac{1}{\beta }}{L}_j{\partial }_{x_2}u=-{\displaystyle \frac{1}{\alpha \beta }}{L}_k{\partial }_{x_3}u,\end{array}$$

(22a)

$$\begin{array}{c}\hfill {\partial }_{x_0}u={\partial }_{x_1}u{\displaystyle \frac{1}{\alpha }}{R}_i={\partial }_{x_2}u{\displaystyle \frac{1}{\beta }}{R}_j={\partial }_{x_3}u\left(-{\displaystyle \frac{1}{\alpha \beta }}\right)R_k.\end{array}$$

(22b)

This form is convenient for symbolic manipulation and for verifying closure under multiplication by constants on the appropriate side.

The matrix-linearized form of the left Cauchy–Riemann relations is summarized in Figure 2.

3.2. CR → Dirac → Laplacian Pipeline and Harmonicity

Recall- the Dirac-type operators (15) and their scalar factorization (16). Applying $\overline{\mathcal{D}}$ to (19) and using (16) yields ${\Delta }_{\alpha ,\beta }f=0$ in the sense that each component is ${\Delta }_{\alpha ,\beta }$-harmonic.

The differentiability–CR–Dirac–Laplacian pipeline is summarized in Figure 3.

Theorem 2

(Harmonicity of components). Let $f\in {\mathcal{C}}^2(U,{\mathbb{H}}_{\alpha ,\beta })$ be left or right differentiable on $U\setminus \mathcal{K}$ and satisfy the corresponding CR system there. Then, each component $u_r$ is ${\Delta }_{\alpha ,\beta }$-harmonic:

$${\Delta }_{\alpha ,\beta }{u}_r={\partial }_{x_0}^2{u}_r-\alpha {\partial }_{x_1}^2{u}_r-\beta {\partial }_{x_2}^2{u}_r+\alpha \beta {\partial }_{x_3}^2{u}_r=0,r=0,1,2,3.$$

• Two Laplace-type operators

Alongside the scalar operator ${\Delta }_{\alpha ,\beta }$ from (16), the CR systems (20) and (21) imply that $(u_0,u_1)$ and $(u_2,u_3)$ separately satisfy diagonal second-order equations governed by

$${\Delta }_1:={\partial }_{x_0}^2-\alpha {\partial }_{x_1}^2+\beta {\partial }_{x_2}^2-\alpha \beta {\partial }_{x_3}^2,{\Delta }_2:={\partial }_{x_0}^2-\alpha {\partial }_{x_1}^2-\beta {\partial }_{x_2}^2+\alpha \beta {\partial }_{x_3}^2.$$

Theorem 3

(Two Laplace-type equations). Let $f\in {\mathcal{C}}^2(U,{\mathbb{H}}_{\alpha ,\beta })$ be left or right differentiable on $U\setminus \mathcal{K}$ and satisfy the corresponding CR system. Then,

$${\Delta }_1{u}_0={\Delta }_1{u}_1=0,{\Delta }_2{u}_2={\Delta }_2{u}_3=0.$$

In particular, each component is also ${\Delta }_{\alpha ,\beta }$-harmonic by the Dirac factorization (16).

Proof sketch.

Differentiate the CR identities (20b)–(20d), combine the resulting second derivatives with weights $(1,-\alpha ,+\beta ,-\alpha \beta )$ for $({\partial }_{x_0}^2,{\partial }_{x_1}^2,{\partial }_{x_2}^2,{\partial }_{x_3}^2)$ in the pairs $(u_0,u_1)$ and $(1,-\alpha ,-\beta ,+\alpha \beta )$ for the pair $(u_2,u_3)$, and use symmetry of mixed partials. Details are given in Appendix D. □

Lemma 1.

The principal symbols are

$$\begin{array}{cc}\hfill \sigma \left({\Delta }_{\alpha ,\beta }\right)\left(\xi \right)& ={\xi }^{\top }{G}_{\alpha ,\beta }\xi ={\xi }_0^2-\alpha {\xi }_1^2-\beta {\xi }_2^2+\alpha \beta {\xi }_3^2,\hfill \\ \hfill \sigma \left({\Delta }_1\right)\left(\xi \right)& ={\xi }_0^2-\alpha {\xi }_1^2+\beta {\xi }_2^2-\alpha \beta {\xi }_3^2,\hfill \\ \hfill \sigma \left({\Delta }_2\right)\left(\xi \right)& ={\xi }_0^2-\alpha {\xi }_1^2-\beta {\xi }_2^2+\alpha \beta {\xi }_3^2.\hfill \end{array}$$

Hence, in general, one has ${\Delta }_1\ne {\Delta }_{\alpha ,\beta }\ne {\Delta }_2$. In the elliptic case, $\alpha =\beta =-1$; all three reduce to the Euclidean Laplacian.

We henceforth refer to Theorem 3 as Theorem 2′, while keeping Theorem 2 (${\Delta }_{\alpha ,\beta }$-harmonicity) for the Dirac pipeline perspective. Proposition 1, Corollary 2, and Theorem 4 remain valid; we add inline notes clarifying the role of ${\Delta }_1$ and ${\Delta }_2$ where appropriate.

Proposition 1

(Principal symbol and regimes). The symbol of ${\Delta }_{\alpha ,\beta }$ is $\sigma \left(\xi \right)={\xi }_0^2-\alpha {\xi }_1^2-\beta {\xi }_2^2+\alpha \beta {\xi }_3^2$. Hence, ${\Delta }_{\alpha ,\beta }$ is elliptic when $\alpha <0$ and $\beta <0$ (Hamiltonian case) and of split signature $(2,2)$ when $\alpha \beta <0$ (coquaternionic case).

Corollary 2

(Regularity in the elliptic regime). If $\alpha =\beta =-1$ and $f\in {\mathcal{C}}^1$ satisfies the CR system on U, then $u_r\in {\mathcal{C}}^{\infty }\left(U\right)$—and, in fact, $u_r$—is real-analytic (interior elliptic regularity).

3.3. A Model Boundary Value Problem (Elliptic Case)

Let $\alpha =\beta =-1$. Denote $B=\{x\in {\mathbb{R}}^4:\parallel x\parallel <1\}$ and $S^3=\partial B$.

Theorem 4

(Poisson representation for components). Let $\alpha =\beta =-1$. Denote $B=\{x\in {\mathbb{R}}^4:\parallel x\parallel <1\}$ and $S^3=\partial B$. If f is left or right differentiable on B and continuous on $\overline{B}$, then each component $u_r$ is harmonic on B with boundary trace $u_r{|}_{S^3}$ and admits

$$\begin{array}{c}\hfill u_r\left(x\right)={\int }_{S^3}P(x,\xi )u_r\left(\xi \right)d\sigma \left(\xi \right),x\in B,\end{array}$$

(23)

where $P(x,\xi )={\displaystyle \frac{1-{\parallel x\parallel }^2}{{\parallel x-\xi \parallel }^4}}$ is the standard Poisson kernel on the unit ball in ${\mathbb{R}}^4$ (see Appendix E for normalization and a brief justification). In particular, the mean value and maximum principles hold component-wise.

3.4. Right–Left Symmetries, Composition, and Computational Checks

• Symmetries and closures.

The matrix relations (22a) and (22b) imply that left CR functions are closed under addition and left multiplication by constants in ${\mathbb{H}}_{\alpha ,\beta }$ (and analogously for right CR under right multiplication). Chain rules require care due to non-commutativity; clean closure holds for affine maps and for compositions with real-valued smooth functions through component-wise operations.

• Hamiltonian specialization

For $(\alpha ,\beta )=(-1,-1)$, the CR systems coincide and

$${\Delta }_{\alpha ,\beta }=\sum _{r=0}^3{\partial }_{x_r}^2.$$

Thus, left/right differentiability implies the ${\mathbb{R}}^4$ harmonicity of all components.

• Split quaternions

For $(\alpha ,\beta )=(-1,1)$,

$${\Delta }_{\alpha ,\beta }u={\partial }_{x_0}^{2}u+{\partial }_{x_1}^{2}u-{\partial }_{x_2}^{2}u-{\partial }_{x_3}^{2}u,$$

so components solve a $(2,2)$ signature equation. The mixed elliptic–hyperbolic nature is reflected in the CR constraints.

• Affine CR families

Let $A_{\mu }\in {\mathbb{H}}_{\alpha ,\beta }$ be constant and set $f\left(x\right)=A_0{x}_0+A_1{x}_1+A_2{x}_2+A_3{x}_3$. Then,

$${\partial }_{x_r}f=A_r,(20)\mathrm{holds}\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{A}_0=\frac{i}{\alpha }{A}_1=\frac{j}{\beta }{A}_2=-\frac{k}{\alpha \beta }{A}_3,$$

producing an explicit affine family of left CR solutions.

• Quadratic map $p\mapsto p^2$

Expansion via the bilinear product (5) shows that when $(\alpha ,\beta )=(-1,-1)$, the components satisfy (20) and (21) and are harmonic; for $(\alpha ,\beta )=(-1,1)$, they satisfy the split relations and solve the corresponding $(2,2)$ Laplace equation.

4. Extensions and Outlook

• Boundary integrals and Fueter-type formulas: Construct Cauchy-type integrals adapted to ${\mathbb{H}}_{\alpha ,\beta }$ and derive Borel–Pompeiu representations that couple the CR systems to boundary data.
• Green kernels in split signature: Develop fundamental solutions for ${\Delta }_{\alpha ,\beta }$ when $\alpha \beta <0$ and analyze radiation/causality conditions.
• Weak formulations: Pose the CR systems on Sobolev scales, establish trace/extension theorems, and study Fredholm properties under boundary conditions.
• Structure-preserving numerics: Design finite/boundary element schemes that encode the linearizations (22a) and (22b) and preserve $G_{\alpha ,\beta }$ symmetries.
• Links to Clifford/Fueter analysis: Connect with Fueter maps and iterated Dirac systems and compare growth/regularity classes.

5. Conclusions

We provided a unified derivation of left/right Cauchy–Riemann (CR) systems in ${\mathbb{H}}_{\alpha ,\beta }$, a sufficiency theorem for differentiability away from the null cone, and a Dirac–factorized second-order theory yielding harmonic components. In the elliptic case $(\alpha ,\beta )=(-1,-1)$, the framework recovers classical Poisson representations, while in the split regime $\alpha \beta <0$, it clarifies the mixed (hyperbolic) character of the component equations. Real $4\times 4$ matrix linearizations furnish a compact calculus well suited for analysis, symbolic verification, and computation.

• Summary of Contributions

• Differentiability via CR systems; Starting from difference quotients, we derived complete left/right CR systems and proved a sufficiency result: under ${\mathcal{C}}^1$ regularity on $U\setminus \mathcal{K}$, the CR identities imply the existence of the corresponding left/right derivative. This establishes a precise quaternionic analogue of the one-dimensional calculus paradigm.
• Second-order potential theory: Using Dirac/Fueter operators $\mathcal{D},\overline{\mathcal{D}}$, we factored the constant coefficient operator

  $${\Delta }_{\alpha ,\beta }={\partial }_{x_0}^2-\alpha {\partial }_{x_1}^2-\beta {\partial }_{x_2}^2+\alpha \beta {\partial }_{x_3}^2=\overline{\mathcal{D}}\mathcal{D}=\mathcal{D}\overline{\mathcal{D}},$$

  and showed that each real component is ${\Delta }_{\alpha ,\beta }$-harmonic. The principal symbol neatly separates the elliptic (Hamiltonian) and split (coquaternionic) regimes.
• Poisson representation (elliptic case): For $(\alpha ,\beta )=(-1,-1)$, we gave a Poisson–type formula on the unit ball in ${\mathbb{R}}^4$, recovering mean-value and maximum principles for the components.
• Matrix linearization: Real $4\times 4$ models of left/right multiplication compress the CR systems into Jacobian matrix identities, enabling direct symbolic checks, stable discretizations, and fast prototyping.
• Examples and contrasts: Polynomial/affine families and split-signature examples illustrate the theory and the role of signature, including the behavior across the null cone.

Scope and limitations. Our arguments require ${\mathcal{C}}^1$ regularity and exclude the degeneracy set $\mathcal{K}=\left\{N\right(p)=0\}$, where invertibility and the M–metric fail. A finer analysis near $\mathcal{K}$ (removable singularities, unique continuation, propagation phenomena in $(2,2)$ signature) remains open. Non-commutativity enforces a left/right dichotomy; sharpening product/chain rules and composition operators beyond the smooth setting is a natural direction.

Outlook. This framework suggests several concrete extensions:

• Boundary integral and Borel–Pompeiu formulas adapted to ${\mathbb{H}}_{\alpha ,\beta }$, linking CR data to boundary values.
• Green kernels and layer potentials for ${\Delta }_{\alpha ,\beta }$ in split signature, with radiation/causality conditions.
• Weak formulations on Sobolev scales, trace/extension theory, and Fredholm properties under natural boundary conditions.
• Spectral and semigroup approaches for evolution problems driven by ${\Delta }_{\alpha ,\beta }$.
• Structure-preserving numerics (finite/boundary elements) exploiting the matrix linearizations and $G_{\alpha ,\beta }$ symmetries.
• Links to Clifford/Fueter analysis, including Fueter maps and iterated Dirac systems, where our linearization offers a practical bridge.

In closing, the differential–potential theoretic synthesis developed here clarifies how quaternionic differentiability controls second-order harmonicity across algebraic regimes, and it provides tools—both conceptual and computational—for the further analysis of generalized quaternionic PDEs.