Let be an open set of . A function in is said to be left (right) -analytic in when
where the Dirac D-operator and its adjoint are the first-order systems of
differential operators in defined by and .
If is a simultaneously left and right -analytic function, then is called an -analytic function. If is a (left) -analytic function in , then is called a (left) -entire function.
Since octonions is non-commutative and non-associative, the product of two left -analytic functions and is generally no longer a left -analytic function. Furthermore, if becomes an octonionic constant function, the product is also probably not a left -analytic function; that is, the collection of left -analytic functions is not a right module (see  ).
The purpose of this paper is to study the analyticity for the product of two left -analytic functions in the framework of complexification of , . Especially, the analyticity for the product of left -analytic functions and constants will be consider more by us.
The rest of this paper is organized as follows. Section 2 is an overview of some basic facts concerning octonions and octonionic analysis. Section 3 we give some sufficient conditions for the product of two left -analytic functions and is also a left -analytic function. In Section 3, we prove that, is a left -analytic function for any constants if and only if is a complex Stein-Weiss conjugate harmonic system. This gives the solution of the problem in  . In the last section we give some applications for our results.
2. Preliminaries: Octonions and Octonionic Analysis
It is well known that there are only four normed division algebras    : the real numbers , complex numbers , quaternions and octonions , with the relations . In other words, for any , , if we define a product “ ” such that and
, where , then the only four values of are 1,2,4,8.
Quaternions is not commutative and octonions is neither commutative nor associative. Unlike , and , the non-associative octonions can not be embedded into the associative Clifford algebras  .
Octonions stand at the crossroads of many interesting fields of mathematics, they have close relations with Clifford algebras, spinors, Bott periodicity, Projection and Lorentzian geometry, Jordan algebras, and exceptional Lie groups, and also, they have many applications in quantum logic, special relativity and supersymmetry   .
Denote the set by
Then the multiplication rules between the basis on octonions are given by   :
and for any triple ,
For each , is called the scalar part of x and is termed its vector part. Then the norm of x is and its conjugate is defined by . We have , Hence, is the inverse of .
Let , then
where is the inner product of vectors and
is the cross product of vectors , with
For any , the inner product and cross product of their vector parts satisfy the following rules  :
We usually utilize associator as an useful tool on ontonions since its non- associativity. Define the associator of any by .
The octonions obey the following some weakened associative laws.
For any , we have (see  )
and the so-called Moufang identities 
Proposition 2.1 (  ). For any , or or .
Proposition 2.2 (  ). Let be three different elements of and . Then .
Since octonions is an alternative algebra (see    ), we have the following power-associativity of octonions.
Proposition 2.3. Let , be elements out of
repetitions being allowed and let be the product of octonions in a fixed associative order . Then is
independent of the associative order , where the sum runs over all distinguishable permutations of
Proof. Let , then is just the coefficient of in the product of . By induction and (2.2), one can easily prove that is independent of the associative order for any . Hence is also independent of the associative order . ,
is called a Stein-Weiss conjugate harmonic system if they satisfy the following equations (see  ):
It is easy to see that if is a Stein-Weiss conjugate harmonic system in an open set of , then there exists a real- valued harmonic function in such that F is the gradient of . Thus is an -analytic function. But inversely, this is not true  .
Example. Observe the -analytic function . Since
is not a Stein-Weiss conjugate harmonic system.
In  Li and Peng proved the octonionic analogue of the classical Taylor theorem. Taking account of Proposition 2.3, we obtain an improving of Taylor type theorem for -analytic functions (see   ).
Theorem A (Taylor). If is a left -analytic function in which containing the origin, then it can be developed into Taylor series
and if is a right -analytic function, then the Taylor series of at the origin is given by
where runs over all possible combinations of elements out of repetitions being allowed.
The polynomials of order in Theorem A is defined by
where the sum runs over all distinguishable permutations of and .
We have the following uniqueness theorem for -analytic functions  .
Proposition 2.4. If is left (right) -analytic in an open connect set and vanishes in the open set , then is identically zero in .
Proof. Without loss of generality, we let which containing the origin and let . Then can be developed into Taylor series
Thus we have
By the uniqueness of the Taylor series for the real analytic function, we have for any and . This shows that is identically zero in and also in . ,
For more references about octonions and octonionic analysis, we refer the reader to   -  .
3. Sufficient Conditions
In what follows we consider the complexification of , it is denoted by .
Thus, is of the form . and are still
called the scalar part and vector part, respectively. The norm of is
and its conjugate is defined by , where is of the
conjugate in the complex numbers. We can easily show that for any , . For any , we may rewrite as , where . The multiplication rules in is the same as in (2.1). Note that is no longer a division algebra. Finally, the properties of associator in (2.2) except that are also true for any :
Example. Let , then
By (3.1) we can get the following lemma, which is useful to deduce our results.
Lemma 3.1. Let and there exists complex numbers and such that or or , then .
For functions, f, under study will be defined in an open set of and
take values in , with the form , where
are the complex-valued functions.
Hence, we say that, a function is left -analytic in an open set of , if and are the left -analytic functions, since
where is the Dirac operator as in Section 1.
In the case of , we call a complex Stein-Weiss conjugate harmonic system, if are the Stein-Weiss conjugate harmonic systems. A left (right) -analytic functions also have the Taylor expansion as in Theorem A.
Now we consider the product of two left -analytic functions in . In general, is no longer left -analytic in . But, in some particular cases, the product can maintain the analyticity for two left -analytic functions and .
Theorem 3.2. Let be two left -analytic functions in . Then is also left -analytic in if satisfy one of the following conditions:
1) or is a complex constant function.
2) is a complex Stein-Weiss conjugate harmonic system in and is an -constant function.
3) is of the form and depend only on and , where are the complex-valued functions.
4) and belong to the following class
5) is of the form , is a constant function, where , and depends only on .
Proof. 1) The proof is trivial.
2) In view of Proposition 2.1 we have when or or for any . Then we have
Since is a complex Stein-Weiss conjugate harmonic system, thus
and for . But , therefore
3) Since are only related to variables and , we have
By Lemma 3.1 it follows that
Thus we get
4) Let and , then we have
By Lemma 3.1 we get
Hence we obtain
5) This case is equivalent to a left quaternionic analytic function right- multiplying by a quaternionic constant, the analyticity is obvious since the multiplication of the quaternion is associative.
The proof of Theorem 3.2 is complete. ,
From Theorem 3.2(d), if , then ; that is, the multiply operation in is closed. Also, the division operation is closed in .
Actually, let , assume , then
Thus we have
An element belongs to is the exponential function:
The results in Theorem 3.2 also hold on octonions(no complexification), since contains . If one switch the locations of , and the “left” change into “right” in Theorem 3.2, then this theorem is also true, since left and right is symmetric. These principles also hold in the rest of this paper.
4. Necessary and Sufficient Conditions
If we consider the product of a left -analytic function and an -constant, we can get the necessary and sufficient conditions for the analyticity(these results obtained in this section for -analytic functions are also described in  ).
Applying Theorem 3.2(a) and (b), if is a left -analytic function and is a complex constant, or is a complex Stein-Weiss conjugate harmonic system and is an -constant, then is a left - analytic function. In what follows we will see that these conditions are also necessary in some sense.
Theorem 4.1. Let , then is a left -analytic function for any left -analytic functions if and only if .
Proof. We only prove the necessity. Taking a left -analytic function , then
Thus . A similar technique yields . Hence . ,
Theorem 4.2. Let . Then for any if and only if is a complex Stein-Weiss conjugate harmonic system in .
Now we postpone the proof of Theorem 4.2 and consider a problem under certain conditions weaker than Theorem 4.2. In  the authors proposed an open problem as follows:
Find the necessary and sufficient conditions for an -valued function , such that the equality holds for any constant .
Note that this problem is of no meaning for an associative system, but octonions is a non-associative algebra, therefore we usually encounter some difficulties while disposing some problems in octonionic analysis. In  the authors added the condition for to study the Cauchy integrals on Lipschitz surfaces in octonions and then prove the analogue of Calderón’s conjecture in octonionic space.
Next we give the answer to the Open Problem as follows.
Theorem 4.3. Let . Then for any if and only if
Proof. By Proposition 2.1, we have
If satisfies (4.1), then .
Inversely, let , and
From Propositions 2.1 and 2.2 we have and when and , respectively. Hence, taking it follows that
Similarly, we take , then
Also we can get
If we require for any constants , from (4.2), (4.3) and (4.4) we obtain
Combining above three equations with the randomicity of we have (4.1) holds. ,
Proof of Theorem 4.2. The sufficient from Theorem 3.2(b). Inversely, if we take in it follows that is a left -analytic function. Thus for any , we have
By Theorem 4.3 we get that satisfies (4.1). On the other hand,
From (4.1) it easily to get , again by (4.5) it follows that
Combining this with (4.1) it shows that is a complex Stein-Weiss conjugate harmonic system in . ,
5. Some Applications and Relations with the C-K Products
From Theorem A we can see that are the basic components for (left) -analytic functions. It is proved in  that the polynomials are all -analytic functions, therefore they are the suitable substitutions of the polynomial in .
Again from Theorem A, since is an item in the Taylor expansion of a left -analytic function, should be also a left analytic function. Applying Theorem 4.2, the conjugate of is probably a Stein-Weiss conjugate harmonic system. The following theorem prove this is true.
Theorem 5.1. For any combination of elements out of repetitions being allowed, is a Stein-Weiss conjugate harmonic system in .
Proof. Let be the appearing times of in . Hence the following equality
shows that is a Stein-Weiss conjugate harmonic system in , where
is a real-valued harmonic function of order with .
Actually, put , the both sides of (5.1) equal to . On
the other hand, is left -analytic in . Thus by Proposition 2.4 we have (5.1) holds. ,
Combining Theorem 3.2(b) and Theorem 5.1 it really shows that all the are left -analytic functions for any . Hence the following series
is a left -analytic function in some open neighborhood of the origin if satisfies certain bounded conditions.
Theorem 5.2. For any combination of k elements out of
repetitions being allowed, let . If ,
then the series (5.2) converges to a left -analytic function in the following region
More over, Particularly, if , then will be a left -entire function.
For any , there exists such that . Thus
From Weierstrass Theorem on octonions  and the analyticity of , then there exists a left -analytic function in such that
and the series uniformly converges to in each compact subset . Again from the expansion of we easily get that .
If , then , since . Therefore is a
left -entire function. ,
Example. Taking for all in (5.2), then
is an -entire function. In fact, (5.3) is the Taylor expansion of the exponential function as in (3.3). From (3.3) we can find satisfies
Corollary 5.3. For any left -analytic function , if the coefficients in its Taylor series about the origin satisfy
Then is a complex Stein-Weiss conjugate harmonic system.
Proof. From (5.4), we easily obtain that all the conjugates of are complex Stein-Weiss conjugate harmonic systems. Hence by Weierstrass Theorem, also is a complex Stein-Weiss conjugate harmonic system in its convergent area. ,
Combining Theorem 3.2(b), Theorems 5.1 and 5.2, by an analogous method in  we can define the Cauchy-Kowalewski product for any two left analytic functions f and g in which containing origin. We let their Taylor expansions be
Then the (left) Cauchy-Kowalewski product of f and g is defined by
where and are the appearing times of i in and , respectively.
We have the following relation for the product and the left Cauchy-Kowalewski product between two left -analytic functions.
Theorem 5.4. Let be two left -analytic functions in which containing origin. If then
Proof. It is easy to see that , then by Proposition 2.4 and the analyticity of and we get . ,
Remark. In this paper we study the analyticity of the product of two left -analytic functions. Theorem 3.2 give some sufficient conditions for the product of two left -analytic functions is also a left -analytic function. From Theorem 5.4 we can see that for two left -analytic functions if and only if this product is just equal to their left Cauchy-Kowalewski product. Since , our result is also true for quaternionic cases.
This work was supported by the Research Project Sponsored by Department of Education of Guangdong Province-Seedling Engineering (NS) (2013LYM0061) and the National Natural Science Foundation of China (11401113).
 Li, X.M., Peng, L.Z. and Qian, T. (2008) Cauchy Integrals on Lipschitz Surfaces in Octonionic Spaces. Journal of Mathematical Analysis and Applications, 343, 763-777.
 Liao, J.Q., Li, X.M. and Wang, J.X. (2010) Orthonormal Basis of the Octonionic Analytic Functions. Journal of Mathematical Analysis and Applications, 366, 335-344.