# Characteristics of Regular Functions Defined on a Semicommutative Subalgebra of 4-Dimensional Complex Matrix Algebra

Authors: Ji Eun Kim
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we give an extended quaternion as a matrix form involving complex components. We introduce a semicommutative subalgebraℂℂ2 of the complex matrix algebra M4,ℂ. We exhibit regular functions defined on a domain in ℂ4 but taking values in ℂℂ2. By using the characteristics of these regular functions, we propose the corresponding Cauchy–Riemann equations. In addition, we demonstrate several properties of these regular functions using these novel Cauchy–Riemann equations. Mathematical Subject Classification is 32G35, 32W50, 32A99, and 11E88.


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## Body

## 1. Introduction

Introduced by Hamilton in 1894, quaternions form an algebra generated as a noncommutative division (associative) algebra. Quaternions are used in physics and engineering fields such as electromechanics, quantum mechanics, and 3D animation (see [1–3]). To increase the utilization of quaternions and expand their application, several researchers have attempted to modify, supplement, or expand quaternions. In particular, considering that quaternions form a number system extending the complex numbers, there have been several studies already made since the past few decades regarding holomorphic functions of a quaternion variable. These researchers have investigated whether the properties of complex functions and the definition of holomorphic functions are applicable to functions of a quaternion variable. To do this, researchers have defined holomorphic (regular) functions of a quaternion variable and they have investigated their properties. Fueter [4] defined regular functions over the quaternion field identified with ℝ4. Based on this definition, Fueter investigated a generalization of the Cauchy–Riemann equations in the complex holomorphic function theory. Delanghe [5] established Stokes’ theorem and Cauchy’s and Green’s formulas for functions with values in Clifford algebra over a quadratic n-dimensional real vector space V with an orthogonal base. Using a generalization of the Cauchy–Riemann equation, Ryan [6] introduced a regularity of quaternion-valued functions and developed a regular function theory of complex Clifford algebra. Sudbery [7] demonstrated basic algebraic properties of quaternions and quaternionic differential forms. Using exterior differential calculus, Sudbery proposed new and simple proofs of most of the main theorems and clarified the relationship between quaternionic analysis and complex analysis. Naser [8] and Nôno [9] introduced certain quaternionic differential operators and defined the hyperholomorphy of quaternionic functions. Nôno and Inenaga [10] developed hyperholomorphic functions of quaternionic variables as holomorphic function theory of ℂ2. Kim et al. [11, 12] researched the properties of regular functions with values in the three-dimensional real skew field (called ternary field T) and reduced quaternions using Clifford analysis. They proposed the corresponding Cauchy–Riemann equations with applications defined on T. Sommen [13] constructed kernels and monogenic and holomorphic functions, leading to connections between the theory of holomorphic functions of several variables and the theory of monogenic functions. Based on these studies, in this paper, we propose a form distinct from the previously attempted expansion of quaternions. In previous studies, extensions of the quaternionic number system and various combinations of quaternions were suggested. Baez [14] applied octonions to spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and exceptional Lie groups. Baez also dealt with their applications in quantum logic, special relativity, and supersymmetry. Imaeda K and Imaeda M [15] presented a 16-dimensional Cayley–Dickson algebra and found its algebraic properties, zero-divisors, and solutions to a general linear equation. Tian [16] provided a complete investigation of real matrix representations of octonions and considered their various applications to octonions, based on the fact that the octonion is an extension of the quaternion by the Cayley–Dickson construction. Gotô and Nôno [17] established a commutative algebra ℂℂ2 identified with ℂ as the subalgebra of the four-dimensional real matrix algebra M4,ℝ. They provided a regularity and properties of the functions of two complex variables with values in ℂℂ2. Rudin [18] studied the integral formulas of the holomorphic functions of several complex variables and presented a function theory of including the boundary behavior of complex functions, complex-tangential phenomena, and quantitative theorems about zero-varieties in the unit ball of ℂn.In this paper, we give algebra using the matrix as the basis by mapping the basis of the quaternion to matrix form. We define an algebraic system based on a matrix with a matrix as a component. This can be expressed as a matrix with complex numbers as components while retaining the characteristics of the basis for generating quaternions. Furthermore, a4×4 matrix is constructed with a matrix that is isomorphic to the basis of quaternions. The algebra introduced in this paper has eight bases, some of which are commutative for products and some are noncommutative. Therefore, in this paper, a function is defined on an algebra generated by a semicommutative basis. Regularity of functions and properties of these regular functions is investigated. In addition, the corresponding Cauchy–Riemann equations, which can be an alternative of the definition of holomorphic function in complex analysis, are derived for the algebra.We develop the foundation of a function theory over a subalgebra of four-dimensional matrix algebra to replace the basis of the quaternion. In Section2, we present preliminaries and notations required for the theory presented in this paper. Sections 3 and 4 present the definitions and propositions related to the semicommutative subalgebra ℂℂ2 of the complex matrix algebra M4,ℂ, and we provide a regularity of functions defined on a domain in ℂ4 with values in ℂℂ2. From the notion of regularity over ℂℂ2, we propose corresponding Cauchy–Riemann equations and several properties of regular functions in ℂℂ2. Finally, in Section 5, the conclusions of this paper are presented.


## 2. Preliminaries and Notations

In this section, we give notations that are needed to prove our main results. LetMn,F be the ring of all n×n matrices over the field F. For our purposes, we would be interested in the case where n=2 and F=ℂ. Put(1)e0=1001,e1=i00−i,e2=01−10,e3=0ii0.These have the following rules:(2)e02=e0,e12=e22=e32=−e0,e1e2=e3=−e2e1,e2e3=e1=−e3e2,e3e1=e2=−e1e3.LetS be a set of matrices defined as follows:(3)S=∑r=03xrer|xr∈ℝr=0,1,2,3.From the rules oferr=0,1,2,3, we note that S is a noncommutative subalgebra of M2,ℂ. By using the matrices err=0,1,2,3, we define four 4×4 matrices:(4)ε0=e000e0,ε1=e100−e1,ε2=e000−e0,ε3=e100e1,ε4=0e2−e20,ε5=0e3e30,ε6=0e2e20,ε7=0e3−e30,where 0 is the 2×2 null matrix. Then, we have the following properties:(5)ε02=ε22=ε42=ε62=ε0,ε12=ε32=ε52=ε72=−ε0,and(6)ε1ε2=ε3=ε2ε1,ε1ε6=ε7=ε6ε1,ε5ε4=ε1=ε4ε5,ε2ε5=ε7=−ε5ε2,ε5ε3=ε6=−ε3ε5,ε3ε4=ε7=−ε4ε3,ε6ε4=ε2=−ε4ε6.Now, let the setS4,ℂ be a subalgebra of M4,ℂ, written as(7)S4,ℂ=z=∑r=07xrεr|xr∈ℝ,r=0,1,…,7,Here, we deal with ε0 as 1. The element z of S4,ℂ is also written as(8)z=x0+x1ε1+x2ε2+x3ε3+x4+x5ε1+x6ε2+x7ε3ε4=x0+x1ε1+x2+x3ε1ε2+x4+x5ε1+x6+x7ε1ε2ε4.From the above expressions, by simply making the following substitutions, we put(9)ζ0=x0+x1ε1,ζ1=x2+x3ε1,ζ2=x4+x5ε1,ζ3=x6+x7ε1,and we obtain(10)z0=ζ0+ζ1ε2,z1=ζ2+ζ3ε2,z¯0=ζ¯0+ζ¯1ε2,z¯1=ζ¯2+ζ¯3ε2,z˜0=ζ0−ζ1ε2,z˜1=ζ2−ζ3ε2,where z¯tt=0,1 denotes the complex conjugate of the complex components of ε1 and z˜t is a changing symbol of a part of the basis of ε2. Then, the elements of S4,ℂ can be represented in the form z0+z1ε4. Let ℂℂ2 be the set S4,ℂ, which is the set of elements of S4,ℂ expressed in the form of z0+z1ε4, denoted by(11)ℂℂ2=z=z0+z1ε4|z0,z1∈M4,ℂ.In addition, the conjugate ofz in ℂℂ2 is defined as z=z¯0+z¯1ε4. Explicitly, we have(12)z¯=x0−x1ε1+x2ε2−x3ε3+x4ε4−x5ε5+x6ε6−x7ε7=x0−x1ε1+x2ε2−x3ε3+x4−x5ε1+x6ε2−x7ε3ε4=x0−x1ε1+x2−x3ε1ε2+x4−x5ε1+x6−x7ε1ε2ε4=ζ¯0+ζ¯1ε2+ζ¯2+ζ¯3ε2ε4=z¯0+z¯1ε4.Forz,w∈ℂℂ2, we can write z=z0+z1ε4 and w=w0+w1ε4, then owing to the identities(13)z0ε4=ε4z˜0,z1ε4=ε4z˜1.Thus, we obtain(14)zw=z0w0+z1w˜1+z0w1+z1w˜0ε4.Sinceε1ε2=ε2ε1, we have(15)zrwt=wtzr,r,t=0,1.However,(16)zrε4=ε4z˜r,wtε4=ε4w˜t.Although, as a result,zw≠wz, it is equal to a part of zw and wz; hence, ℂℂ2 is a semicommutative subalgebra of M4,ℂ. We first obtain an expression for z∗ in order to define z. Let z∗=zt¯, where zt is the transposed matrix of z. Explicitly, z∗ (as an element of ℂℂ2) is given by(17)z∗=zt¯=x0−x1ε1+x2ε2−x3ε3+x4ε4−x5ε5−x6ε6+x7ε7=x0−x1ε1+x2−x3ε1ε2+x4−x5ε1−x6−x7ε1ε2ε4=ζ¯0+ζ¯1ε2+ζ¯2−ζ¯3ε2ε4=z¯0+z¯˜1ε4.From the above results, the normz of z is therefore given by(18)z=12trzz∗1/2=∑j=07xj21/2,where trz is the trace of matrix zz∗.Proposition 1.
Forz,w∈ℂℂ2, the following properties for the norm z are satisfied:(1)
z+w≤z+w(2)
zw≤zw,z,w∈ℂℂ2
We now define differential operators as algebraic expressions for the elemental form ofℂℂ2. The differential operator here includes elements in the form of a matrix. We consider the following differential operators over ℂℂ2:(19)D˜1=12∂∂z˜0−ε4∂∂z1,D1=12∂∂z0+ε4∂∂z1,D˜2=12∂∂z0−ε4∂∂z1,D2=12∂∂z˜0+ε4∂∂z1,D˜3=12∂∂z˜0−ε4∂∂z˜1,D3=12∂∂z0+ε4∂∂z˜1,D˜4=12∂∂z0−ε4∂∂z˜1,D4=12∂∂z˜0+ε4∂∂z˜1,∂∂z0=12∂∂ζ0+ε2∂∂ζ1,∂∂z1=12∂∂ζ2+ε2∂∂ζ3,∂∂z˜0=12∂∂ζ0−ε2∂∂ζ1,∂∂z˜1=12∂∂ζ2−ε2∂∂ζ3,where ∂/∂ζjj=0,1,2,3 has the same pattern operator form as usual complex differential operators.As a concrete example, we have(20)∂∂ζ0ζ0=12∂∂x0−ε1∂∂x1x0+x1ε1=12∂∂x0−i∂∂x10000∂∂x0+i∂∂x10000∂∂x0+i∂∂x10000∂∂x0−i∂∂x1x0+ix10000x0−ix10000x0−ix10000x0+ix1=1000010000100001.In this case, let the4×4 identity matrix be 1. If the calculation is performed by expressing it as a matrix, the following results are obtained:(21)∂∂z0z0=1,∂∂z˜0z0=0.Next, letΩ be a domain in ℂ4. We consider a function f:Ω⟶ℂℂ2 by(22)z=z0,z1↦fz=f0z0,z1+f1z0,z1ε4,where f0 and f1 have the form φ+ψε2 with φ and ψ being complex-valued functions. For example, for a function fz=z2, this takes the form fz=z02+z1z˜1+z0z1+z1z˜0ε4, denoted by f0=z02+z1z˜1 and f1=z0z1+z1z˜0.The differential operatorD˜kk=1,2,3,4 operates on f as follows: in the case of n=1, applying the differential operator from the left, we have(23)D˜1f=12∂∂z˜0−ε4∂∂z1f0+f1ε4=12∂f0∂z˜0−∂f˜1∂z˜1+∂f1∂z˜0−∂f˜0∂z˜1ε4,and applying the differential operator from the right, we obtain(24)fD˜1=12f0+f1ε4∂∂z˜0−ε4∂∂z1=12∂f0∂z˜0−∂f1∂z1+−∂f0∂z˜1+∂f1∂z0ε4.Now, let us define the regular functions to be applied inℂℂ2 by using the differential operator D˜kDkk=1,2,3,4 for functions defined in ℂℂ2.


## 3. Definitions and Properties of Regularity

Definition 1.
LetΩ be a domain in ℂ4. Let f=f0+f1ε4 be a function defined on Ω. The function f is said to be JLkJRk-regular function k=1,2,3,4 in Ω if(1)
f0 and f1 are separately continuously holomorphic functions defined on Ω(2)
D˜kf=0,fD˜k=0 in Ω, where k=1,2,3,4
When a functionf is JLkJRk-regular k=1,2,3,4, it means that at least the above definition is satisfied for at least one of k=1,2,3,4.Remark 1.
Regardingf0 and f1 being separately continuous holomorphic functions, we mean that f0=φ1+φ2ε2 and f1=ψ1+ψ2ε2 such that the components φr and ψrr=1,2 of each ftt=0,1 are holomorphic complex-valued functions.Lemma 1 (Cauchy–Riemann equation inℂℂ2).
Condition 2 of Definition1 for JLk-regular function is equivalent to(25)∂f0∂z˜0=∂f˜1∂z˜1∂f1∂z˜0=∂f˜0∂z˜1,ifk=1,∂f0∂z0=∂f˜1∂z˜1,∂f1∂z0=∂f˜0∂z˜1,ifk=2,∂f0∂z˜0=∂f˜1∂z1,∂f1∂z˜0=∂f˜0∂z1,ifk=3,∂f0∂z0=∂f˜1∂z1,∂f1∂z0=∂f˜0∂z1,ifk=4.
In addition, forJRk-regular function, Condition 2 is equivalent to(26)∂f0∂z˜0=∂f1∂z1,∂f1∂z0=∂f0∂z˜1,ifk=1,∂f0∂z0=∂f1∂z1,∂f1∂z˜0=∂f˜0∂z˜1,ifk=2,∂f0∂z˜0=∂f1∂z˜1,∂f1∂z0=∂f0∂z1,ifk=3,∂f0∂z0=∂f1∂z˜1,∂f1∂z˜0=∂f0∂z1,ifk=4.Theorem 1.
LetΩ be an open set in ℂ4, and let f be a regular function on Ω. Then, the derivative f′ of f, also denoted as Df, is defined as(27)f′=∂f∂x0=−i^∂f∂z0.on Ω.Proof.
From the definition ofJLk-regular function defined on ℂℂ2, in case k=1, Condition 2 can be expressed as follows:(28)D˜1f=12∂f0∂z˜0−∂f˜1∂z˜1+∂f1∂z˜0−∂f˜0∂z˜1ε4=0.
Sincef satisfies(29)∂f0∂z˜0−∂f˜1∂z˜1=0,∂f1∂z˜0−∂f˜0∂z˜1=0,we obtain(30)∂f0∂z˜0=∂f˜1∂z˜1,∂f1∂z˜0=∂f˜0∂z˜1.
Similarly, we can derive the Cauchy–Riemann equations for the other cases.
For example, letf be an identity function such that fz=z. The function f is JLk-regular k=1,2,3,4 defined on ℂℂ2. Let f be a function such that fz=z2 be JL1-regular defined on ℂℂ2. Let f be JLkJRk-regular functions k=1,2,3,4 defined in Ω. Then, we propose the JLkJRk-derivative DkffDkk=1,2,3,4 of f and find the following theorem.Theorem 2.
LetΩ be a domain in ℂ4. Let f be a JLkJRk-regular function defined in Ω. Then, the derivative DkffDkk=1,2,3,4 of f satisfies(31)Dkf=fDk=∂f∂ζ0,k=1,2,3,4.Proof.
Sincef is JL1-regular defined on Ω, the function f satisfies D˜1f=0. Hence, we have(32)D1f=12∂f0∂z0+∂f˜1∂z˜1+∂f1∂z0−∂f˜0∂z˜1ε4=12∂f0∂z0+∂f0∂z˜0+∂f1∂z0+∂f1∂z˜0ε4=12∂f∂z0+∂f∂z˜0=∂f∂ζ0.
Whenk=2,3,4, by using the equation D˜kf=0, we obtain the same results, that is,(33)D2f=D3f=D4f=12∂f∂z0+∂f∂z˜0=∂f∂ζ0.
Similarly, whenf is JRk-regular k=1,2,3,4 defined on Ω, then by the equation fD˜k=0k=1,2,3,4, we obtain(34)fDk=12∂f∂z0+∂f∂z˜0,k=1,2,3,4.
Thus, theJLkJRk-derivative of f satisfies(35)Dkf=fDk=∂f∂ζ0,k=1,2,3,4.
From Theorem1, for a separately continuous holomorphic JLkJRk-regular function fk=1,2,3,4 defined in Ω, we denote(36)Dknfz=fDknz=∂n∂ζ0nfz=fz∂n∂ζ0n.


## 4. Properties ofJLkJRk-Regular ( k=1,2,3,4 Functions

In this section, letΩ be a domain in ℂ4. Based on the definition and properties of JLk-regularity k=1,2,3,4, we determine whether the properties of holomorphic function of complex variables extend JLk-regular function k=1,2,3,4 defined on Ω. Suppose that Ω1,Ω2⊂ℂ4 are two open neighborhoods; then, the variables in Ω1 can be written as z=z0,z1 and Ω2 can be written as w=w0,w1. Any mapping g:Ω1⟶Ω2 can be described by(37)w0=g0z0,z1,w1=g1z0,z1.The mappingg is called a JLk-regular mapping k=1,2,3,4 if the functions g0 and g1 are JLk-regular functions k=1,2,3,4 in Ω1. If fw0,w1=fw is any function defined in Ω2, then composition fg is a well-defined function in Ω1. If fw is a JLk-regular function k=1,2,3,4 in Ω2 and if g:Ω1⟶Ω2 is a JLk-regular mapping k=1,2,3,4, then the composition w=fg is a JLk-regular function k=1,2,3,4 in Ω1.Next, the following quadratic form is factors that play an important role in the integral calculus ofJLk-regular functions k=1,2,3,4:(38)ω1=dz0∧dz1∧dz˜1−dz0∧dz˜0∧dz1ε4,ω2=dz˜0∧dz1∧dz¯1+dz0∧dz˜0∧dz1ε4,ω3=dz0∧dz1∧dz˜1+dz0∧dz˜0∧dz˜1ε4,ω4=dz˜0∧dz1∧dz˜1−dz0∧dz˜0∧dz˜1ε4.Furthermore, we note that(39)dzi∧dzj=−dzj∧dzi,dzi∧dzi=0,i,j=0,1,dΦ:=dz0∧dz˜0∧dz1∧dz˜1.Definition 2 (Morera’s theorem inℂℂ2).
Let a functionf be continuous in Ω. Suppose the integral of ωkfk=1,2,3,4 over the boundary ∂U of any sphere U in Ω be equal to zero. Then, f is a JLk-regular function k=1,2,3,4 in Ω.Proof.
We assume that the functionf is smooth in Ω. Let U be a sphere in Ω. Based on this hypothesis, we have(40)∫∂Uωkf=0,k=1,2,3,4.
From Stokes’ formula in complex analysis, we obtain(41)∫Udkωkf=∫∂Uωkf=0,k=1,2,3,4.
Hence, fork=1, the integral can be calculated as follows:(42)∫U∂f0∂z˜0−∂f˜1∂z˜1dΦ=∫U∂f1∂z˜0+∂f˜0∂z˜1dΦ=0.
Due to the smoothness off, the expression under the integral sign is continuous. Since the fact U is an arbitrary sphere in Ω, it follows that(43)∂f0∂z˜0=∂f˜1∂z˜1,∂f1∂z˜0=−∂f˜0∂z˜1,in Ω. For cases k=2,3,4, we can obtain similar conclusions.
Next, we consider the relationship between theJLk-regular function k=1,2,3,4 and series expansions in ℂℂ2.Lemma 2.
Letf be a homogeneous polynomial of degree n with respect to z0 and z1. If f is a JLk-regular function k=1,2,3,4 in ℂ4, then f satisfies(44)fz=1n!∂nfz∂z0nzn.Proof.
According to the hypothesis, the functionfz is a homogeneous polynomial; thus, f satisfies(45)fz=1n∂fz∂z0z.
Since∂f/∂z0 is a homogeneous polynomial of degree n−1, we have(46)∂fz∂z0=1n−1∂2fz∂z02z.
Continuing this process, we can obtain(47)fz=1n!∂nfz∂z0nzn.Proposition 2.
LetΩ be a domain in ℂ4 and fz be a function defined on a neighborhood V of α∈ℂ4 with values in ℂℂ2. The function f is a JLk-regular function k=1,2,3,4 in Ω and α∈Ω if and only if f has a power series expansion in V such that for z∈V,(48)fz=∑n=0∞anz−αn,where an=1/n!∂nfα/∂z0nn=0,1,2,….Proof.
Without loss of generality, letα=0. Suppose that f is JLk-regular k=1,2,3,4 in Ω with a value in ℂℂ2; then, f is holomorphic in Ω. Thus, there exists a neighborhood V of zero such that for z∈V,(49)fz=∑n=0∞Pnz,where Pnz are homogeneous polynomials of degree n with respect to z0 and z1. Since the above series converges uniformly in V and Pnz are homogeneous polynomials of degree n with respect to z0 and z1, when k=1, we have(50)D1∗fz=12∂∂z¯0+ε4∂∂z1fz=∑n=0∞12∂∂z¯0+ε4∂∂z1Pnz=0.
Hence,Pnz is a JLk-regular k=1,2,3,4 in Ω. By Lemma 2, since(51)∂nf0∂z0n=∂n∂z0n∑n=0∞Pnz,we have(52)fz=∑n=0∞Pnz=∑n=0∞1n!∂nPnz∂z0nzn=∑n=0∞1n!∂nf0∂z0nzn=∑n=0∞anzn,z∈V.
Suppose that(53)fz=∑n=0∞anzn,where an=1/n!∂nf0/∂z0nn=0,1,2,…. Since fz converges uniformly in V, we obtain(54)D1∗fz=12∂∂z¯0+ε4∂∂z1fz=∑n=0∞12an∂∂z¯0+ε4∂∂z1zn=12∑n=0∞an∂∂z¯0z0+z1ε4n+ε4∂∂z1z0+z1ε4n=12∑n=0∞annz0+ε2z1n−1+ε2nz0+ε2z1n−1ε2=0.
Thus,fz is a JLk-regular function k=1,2,3,4 in V.
From Proposition2, equivalently, we can define JLk-regular functions k=1,2,3,4 in Ω which is a domain in ℂ4 as follows.Definition 3.
LetΩ be a domain in ℂ4. A function f=f0+f1ε4 is said to a JLk-regular function k=1,2,3,4 on Ω if every point a∈Ω corresponds to a neighborhood U admitting a power series(55)∑n=0∞anz−αn,which converges to fz for z∈U.
From the above results, the following statements are equivalent to each other:(1)
f is a JLk-regular function k=1,2,3,4 defined on Ω.(2)
For every pointa∈Ω, there is a neighborhood U admitting a power series(56)fz=∑n=0∞anz−αn,z∈U.(3)
For a smooth boundary∂U such that U˜⊂Ω, we have(57)∫∂Uωkf=0,whereωkfk=1,2,3,4 is the quaternion product of the form of the function f=f1+f2ε4.Corollary 1.
LetΩ be a domain in ℂ4 and f be a JLk-regular function k=1,2,3,4. If f is infinitely differentiable in ℂℂ2 and(58)fz=∑n=0∞anz−αn,then we have(59)an=1n!Dknfα=1n!∂n∂z0nfα,n=0,1,2,….


## 5. Conclusion

In this paper, we developed the foundation of a function theory over a subalgebra of four-dimensional matrix algebra as an alternative to the standard basis of quaternion algebra. Furthermore, we introduced a semicommutative subalgebraℂℂ2 of the complex matrix algebra M4,ℂ, and we propose regular functions defined in a domain in ℂ4 with values in ℂℂ2. From the definition of regularity over ℂℂ2, we derived corresponding Cauchy–Riemann equations and several properties of regular functions in ℂℂ2. Throughout this paper, it has been possible to make various attempts to utilize quaternions by demonstrating a series expansion of quaternion variable. Regular functions defined on quaternions can be applied to this extension of the algebra of quaternions introduced in this paper. In addition, it is possible to present the extensibility of formulas derived in this paper based on proposed quaternionic theory for differential operators and series expansion. In the future, we plan to verify properties, such as ∂¯-closed form and regularity over a domain of holomorphy, by using certain regular functions and Cauchy–Riemann equations.



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