#### Abstract

We define a new hypercomplex structure of and a regular function with values in that structure. From the properties of regular functions, we research the exponential function on the reduced quaternion field and represent the corresponding Cauchy-Riemann equations in hypercomplex structures of .

#### 1. Introduction

Meglihzon [1], Sudbery [2], and Fueter [3] demonstrated that there are three possible approaches (the Cauchy approach, Weierstrass approach, and Riemann approach) in the theories of functions that would generalize holomorphic functions with respect to several complex variables. Sudbery [2], Soucek [4], and Sommen [5] attempted to research the Cauchy approach using differential forms and differential operators in Clifford analysis. Fueter [3] and Naser [6] studied the properties of quaternionic differential equations as a generalization of the extended Cauchy-Riemann equations in the complex holomorphic function theory. Nôno [7–9] and Sudbery [2] gave a definition and the development of regular functions over the quaternion field. Ryan [10, 11] developed the theories of regular functions in a complex Clifford analysis using a generalization of the Cauchy-Riemann equation. Malonek [12] considered analogously the function theory of hypercomplex variables. He defined the hypercomplex differentiability for the existence of a function over the Clifford algebra and monogenicity based on a generalized Cauchy-Riemann system. Gotô and Nôno [13] and Koriyama et al. [14] dealt with differential operators with the derivative of regular functions in quaternion.

We shall denote by , , and , respectively, the field of complex numbers, the field of real numbers, and the set of all integers. We [15, 16] showed that any complex-valued harmonic function in a pseudoconvex domain of has a hyperconjugate harmonic function in such that the quaternion-valued function is hyperholomorphic in and gave a regeneration theorem in quaternion analysis in the view of complex and Clifford analysis. Further, we [17, 18] investigated the existence of the hyperconjugate harmonic functions of the octonion number system and some properties of dual quaternion functions.

In this paper, we introduce the Fueter variables on and investigate a hypercomplex structure of . We define regular functions and obtain the representation of the corresponding Cauchy-Riemann equations for regular functions in the reduced quaternion field.

#### 2. Preliminaries

A three-dimensional, noncommutative, and associative real field, called a ternary number system, is constructed by three base elements , , and which satisfy In addition, let be the identity of a ternary number system and identifies the imaginary unit in the complex field, and where and are real variables. They satisfy the equations where , , , and are real variables.

For any two elements and of , their product is given by where the corresponding commutative inner product satisfies and the corresponding noncommutative outer product satisfies The conjugation , the corresponding norm , and the inverse of in are given by

For any element in , we have the corresponding exponential function denoted by

Theorem 1. *Let be an arbitrary number in . Then the corresponding exponential function of in is given as
**
where .**Furthermore, as hyperbolic functions, one has
**
where .*

* Proof. *For any element of ,
Since a scalar part of is , a vector part of is , and , by [19],
and, similarly, we have
Then we have
Also, we obtain
Since (15) has to be equal to (14), , that is, or . Therefore, or , and then or , where . If , then
Similarly, if , then
Further, by the Euler formula and the addition rule of trigonometric functions,
Since and , we have
Since
we obtain
and, similarly,
Since (22) has to be equal to (21), , that is, or . Therefore, or , and then or , where . If , then
Similarly, if , then

*Remark 2. *By Theorem 1 and the properties of the Euler formula, if , then we can write
also, if , then
where and are the conjugate Fueter variables of (see [20]).

Let be an open subset of and let a function be defined by the following form on with values in : satisfying where , and are real-valued functions.

From the chain rule, we use the following differential operators: where in . We have the following equations: and then, the operator operates to as follows: Thus, we have a corresponding Laplacian in the reduced quaternion :

*Remark 3. *Let Ω be an open set of . From the definition of the differential operators in , we have
and, therefore,
Similarly, we have
and, therefore,

*Definition 4. *Let be an open set in and for any element in . A function is said to be* L(R)*-regular on if the following conditions are satisfied:(i) are continuously differential functions on , and(ii) on .

In particular, the equation of Definition 4 is equivalent to Moreover, (38) is equivalent to the following system: The above system is a corresponding Cauchy-Riemann system in .

*Remark 5. *From the multiplications of , the equation of Definition 4 is equivalent to
Also, the above equation (40) is equivalent to the following system:
Further, the above system (41) is also a corresponding Cauchy-Riemann system in . Since the system (39) is equivalent to the system (41), we say that of Definition 4 is a regular function on . When the function is either an* L*-regular function or an* R*-regular function on , we simply say that is a regular function on .

#### 3. Properties of Regular Functions with Values in

We define the derivative of by the following:

Proposition 6. *Let be an open set in and let a function be a regular function defined on . Then
*

* Proof. *From the definition of a regular function , we have
Therefore,
Hence, we obtain the equation
Similarly, by calculating the derivative of ,
Therefore, we have the equation
Further, using the same procedure, we obtain the equations

Proposition 7. *Let be an open set in . If is a regular function on , then we have
**
where is a positive integer.*

* Proof. *Since is a regular function on with values in , by Definition 4,
Hence, is a regular function with values in . From Proposition 6, we have
By repeating the above process, we can obtain the equation

We let on an open set in .

Theorem 8. *Let be an open set in . If is a regular function on , then the following equation holds true:
*

* Proof. *Since is a regular function on , we have the following system:
By the definition of , we have
From Proposition 7, we have . Hence, by calculating and comparing the above polynomials, we obtain that is equal to .

Next, we consider a differential form

Theorem 9. *Let be an open set in and let be any domain on with a smooth distinguished boundary such that . If is a regular function on , then one has
**
where is the reduced quaternionic product of the form on the function .*

* Proof. *Since , we have
where in . From the corresponding Cauchy-Riemann system (39) for in , we have the system (56). Hence, and, therefore, by Stokes theorem, we obtain the following result:

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The third author was supported by a 2-Year Research Grant of Pusan National University.