The Analyticity for the Product of Analytic Functions on Octonions and Its Applications

Given two left c O -analytic functions , f g in some open set Ω of 8 R , we obtain some sufficient conditions for fg is also left c O -analytic in Ω . Moreover, we prove that f λ is a left c O -analytic function for any constants c λ∈O if and only if f is a complex Stein-Weiss conjugate harmonic system. Some applications and connections with CauchyKowalewski product are also considered.


Introduction
Let Ω be an open set of 8 R .A function f in  Since octonions is non-commutative and non-associative, the product ( ) ( ) f x g x of two left O -analytic functions ( ) f x and ( ) g x is generally no longer a left O -analytic function.Furthermore, if ( ) becomes an octonionic constant function, the product ( ) f x λ is also probably not a left O -analytic function; that is, the collection of left O -analytic functions is not a right module (see [1]).
The purpose of this paper is to study the analyticity for the product of two left Especially, the analyticity for the product of left c O -analytic functions and c O 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 ( ) ( ) ( ) f x is a complex Stein-Weiss conjugate harmonic system.This gives the solution of the problem in [2].In the last section we give some applications for our results.

Preliminaries: Octonions and Octonionic Analysis
It is well known that there are only four normed division algebras [3]  Quaternions H is not commutative and octonions O is neither commutative nor associative.Unlike R , C and H , the non-associative octonions can not be embedded into the associative Clifford algebras [6].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 [3] [4].
Denote the set W by  , 0,1, , 7 x is called the scalar part of  is termed its vector part.Then the norm of x is ( ) and its conjugate is defined by is the inverse of ( ) : is the inner product of vectors , x y and ) ( )


For any , x y ∈O , the inner product and cross product of their vector parts satisfy the following rules [8]: We usually utilize associator as an useful tool on ontonions since its nonassociativity.Define the associator [ ] , , x y z of any , , x y z ∈O by The octonions obey the following some weakened associative laws.

, x y z y z x x z y x y z x x y x x y
and the so-called Moufang identities [5] ( ) ( ) e e e = ± ., , , e e e  and ( ) e e e ≠ ± .Then ( ) ( ) e e e e e e = − .
repetitions being allowed and let ( ) be the product of n octonions in a fixed associative order n ⊗ .Then ( ) ( ) independent of the associative order n ⊗ , where the sum runs over all distinguishable permutations of ( ) . By induction and (2.2), one can easily prove that ( ) is independent of the associative order n ⊗ for any x ∈ O .Hence ( ) ( ) is called a Stein-Weiss conjugate harmonic system if they satisfy the following equations (see [11]): ( ) 0 0, 0 .
, , , , , , , then there exists a real- valued harmonic function Φ in Ω such that F is the gradient of Φ .Thus But inversely, this is not true [12].
Example.Observe the O -analytic function ( ) ( ) g is not a Stein-Weiss conjugate harmonic system.
In [13] 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 O -analytic functions (see [14] [15]).
Theorem A (Taylor).If ( ) f x is a left O -analytic function in Ω which containing the origin, then it can be developed into Taylor series analytic function, then the Taylor series of f at the origin is given by runs over all possible combinations of k elements out of { } 1, , 7  repetitions being allowed.
The polynomials where the sum runs over all distinguishable permutations of ( ) Proof.Without loss of generality, we let E which containing the origin and let 0 0 x = .Then f can be developed into Taylor series ( ) By the uniqueness of the Taylor series for the real analytic function, we have ( ) and k ∈ N .This shows that f is identically zero in E and also in Ω . For more references about octonions and octonionic analysis, we refer the reader to [7] [13]-[20].

Sufficient Conditions
In what follows we consider the complexification of O , it is denoted by c O .
and its conjugate is defined by O is the same as in (2.1).Note that c O is no longer a division algebra.Finally, the properties of associator in (2.2) Example.Let O , with the form ( ) ( ) , where ( )( ) Hence, we say that, a function ( ) ( ) ( )  Now we consider the product ( ) ( ) In general, ( ) ( ) But, in some particular cases, the product ( ) ( ) Then ( ) ( ) x g x satisfy one of the following conditions: and ( ) ( ) , f x g x depend only on 0 x and i x , where 0 , i f f are the complex-valued functions.Proof. 1) The proof is trivial.
3) Since ( ) ( ) , f x g x are only related to variables 0 x and i x , we have and ( ) + ∑ , then we have ( ) ( ) ( ) By Lemma 3.1 we get e g e g e e g x x x Hence we obtain This case is equivalent to a left quaternionic analytic function rightmultiplying by a quaternionic constant, the analyticity is obvious since the multiplication of the quaternion is associative.
The proof of Theorem 3.2 is complete. that is, the multiply operation in S is closed.Also, the division operation is closed in S .
Actually, let ( ) ( ) ( ) )( ) An element belongs to S is the exponential function: The results in Theorem 3.2 also hold on octonions(no complexification), since c O contains O .If one switch the locations of ( ) ( ) , f x g x , 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.

Necessary and Sufficient Conditions
If we consider the product of a left c O -analytic function and an c O -constant, we can get the necessary and sufficient conditions for the analyticity(these results obtained in this section for O -analytic functions are also described in [19]).
Applying Theorem 3.2(a) and (b), if ( ) In what follows we will see that these conditions are also necessary in some sense.O -analytic functions f if and only if λ ∈ C .Proof.We only prove the necessity.Taking a left c O -analytic function . 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 [2] the authors added the condition ( ) for ( ) f x 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.
Inversely, let ( ) , , Similarly, we take , , 2 , Combining above three equations with the randomicity of ( ) , , α β γ we have Thus for any c λ ∈O , we have By Theorem 4.3 we get that f satisfies (4.1).On the other hand, ( ) From (4.1) it easily to get 0 f ∇ × = , again by (4.5) it follows that shows that ( ) ( ) is a real-valued harmonic function of order ( ) Actually, put 0 0 x = , the both sides of (5.1) equal to   From Weierstrass Theorem on octonions [13] and the analyticity of ( )

( ) 1 ,
C Ω O is said to be left (right) O -analytic in Ω when

.
D-operator and its adjoint D are the first-order systems of If f is a simultaneously left and right O -analytic function, then f is called an O -analytic function.If f is a (left) O -analytic function in 8 R , then f is called a (left) O -entire function.

cO
-analytic functions in the framework of complexification of O , c O .
x g x of two left c O -analytic functions ( ) f x and ( ) g x is also a left c O -analytic function.In Section 3, we prove that, ( ) f x λ is a left c O -analytic function for any constants c λ ∈O if and only if [4] [5]: the real numbers R , complex numbers C , quaternions H and octonions O , with the relations ⊆ ⊆ ⊆ R C H O .In other words, for any then the only four values of n are 1,2,4,8.

8 Ω
following uniqueness theorem for O -analytic functions [7].Proposition 2.4.If f is left (right) O -analytic in an open connect set ⊂ R and vanishes in the open set the scalar part and vector part, respectively.The norm of c i  is of the conjugate in the complex numbers.We can easily show that for any , rewrite  asx iy = +  , where , x y ∈O .The multiplication rules in c

By ( 3 . 1 ).
we can get the following lemma, which is useful to deduce our results.Lemma 3.1.Let , , For functions, f, under study will be defined in an open set Ω of 8 R and take values in c x are the Stein-Weiss conjugate harmonic systems.A left (right) c O -analytic functions ( ) g x also have the Taylor expansion as in Theorem A.
x g x can maintain the analyticity for two left c O -analytic functions ( ) f x and ( ) g x .Theorem 3.2.Let ( ) ( ) , f x g x be two left c O -analytic functions in Ω .
( ) g x belong to the following class

Theorem 4 . 1 .
Let c λ ∈O , then f λ is a left c O -analytic function for any left c f λ = for any c λ ∈O if and only if f is a complex Stein-Weiss conjugate harmonic system in Ω .Now we postpone the proof of Theorem 4.2 and consider a problem under certain conditions weaker than Theorem 4.2.In [2] the authors proposed an open problem as follows:Find the necessary and sufficient conditions for an c O -valued function f , such that the equality


is left O -analytic in 8 R .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 c O -analytic function in some open neighborhood Λ of the origin if series (5.2) converges to a left c O -analytic function ( ) f x in the following region