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

Given two left Oc-analytic functions f, g in some open set Ω of R8, we obtain some sufficient conditions for fg is also left Oc-analytic in Ω. Moreover, we prove fλ that is a left Oc-analytic function for any constants λOc if and only if is a complex Stein-Weiss conjugate harmonic system. Some applications and connections with Cauchy-Kowalewski product are also considered.

1. Introduction

Let $\Omega$ be an open set of ${R}^{8}$ . A function $f$ in ${C}^{1}\left(\Omega ,O\right)$ is said to be left (right) $O$ -analytic in $\Omega$ when

$Df=\underset{i=0}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\frac{\partial f}{\partial {x}_{i}}=0\text{ }\left(fD=\underset{i=0}{\overset{7}{\sum }}\frac{\partial f}{\partial {x}_{i}}{e}_{i}=0\right),$

where the Dirac D-operator and its adjoint $\stackrel{¯}{D}$ are the first-order systems of

differential operators in ${C}^{1}\left(\Omega ,O\right)$ defined by $D={\sum }_{0}^{7}\text{ }\text{ }{e}_{i}\frac{\partial }{\partial {x}_{i}}$ and $\stackrel{¯}{D}={e}_{0}\frac{\partial }{\partial {x}_{0}}-{\sum }_{1}^{7}\text{ }{e}_{i}\frac{\partial }{\partial {x}_{i}}$ .

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 ${R}^{8}$ , then $f$ is called a (left) $O$ -entire function.

Since octonions is non-commutative and non-associative, the product $f\left(x\right)g\left(x\right)$ of two left $O$ -analytic functions $f\left(x\right)$ and $g\left(x\right)$ is generally no longer a left $O$ -analytic function. Furthermore, if $g\left(x\right)\equiv \lambda$ becomes an octonionic constant function, the product $f\left(x\right)\lambda$ 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 ${O}^{c}$ -analytic functions in the framework of complexification of $O$ , ${O}^{c}$ . Especially, the analyticity for the product of left ${O}^{c}$ -analytic functions and ${O}^{c}$ 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 $f\left(x\right)g\left(x\right)$ of two left ${O}^{c}$ -analytic functions $f\left(x\right)$ and $g\left(x\right)$ is also a left ${O}^{c}$ -analytic function. In Section 3, we prove that, $f\left(x\right)\lambda$ is a left ${O}^{c}$ -analytic function for any constants $\lambda \in {O}^{c}$ if and only if $\stackrel{¯}{f\left(x\right)}$ 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.

2. Preliminaries: Octonions and Octonionic Analysis

It is well known that there are only four normed division algebras [3] [4] [5] : the real numbers $R$ , complex numbers $C$ , quaternions $H$ and octonions $O$ , with the relations $R\subseteq C\subseteq H\subseteq O$ . In other words, for any $x=\left({x}_{1},\cdots ,{x}_{n}\right)$ , $y=\left({y}_{1},\cdots ,{y}_{n}\right)\in {R}^{n}$ , if we define a product “ $xy$ ” such that $xy\in {R}^{n}$ and

$|x\cdot y|=|x||y|$ , where $|x|=\sqrt{{\sum }_{1}^{n}\text{ }{x}_{i}^{2}}$ , then the only four values of $n$ are 1,2,4,8.

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

$W=\left\{\left(1,2,3\right),\left(1,4,5\right),\left(1,7,6\right),\left(2,4,6\right),\left(2,5,7\right),\left(3,4,7\right),\left(3,6,5\right)\right\}.$

Then the multiplication rules between the basis ${e}_{0},{e}_{1},\cdots ,{e}_{7}$ on octonions are given by [3] [7] :

${e}_{0}^{2}={e}_{0},\text{\hspace{0.17em}}{e}_{i}{e}_{0}={e}_{0}{e}_{i}={e}_{i},\text{\hspace{0.17em}}{e}_{i}^{2}=-1,\text{\hspace{0.17em}}i=1,2,\cdots ,7,$

and for any triple $\left(\alpha ,\beta ,\gamma \right)\in W$ ,

${e}_{\alpha }{e}_{\beta }={e}_{\gamma }=-{e}_{\beta }{e}_{\alpha },\text{ }{e}_{\beta }{e}_{\gamma }={e}_{\alpha }=-{e}_{\gamma }{e}_{\beta },\text{ }{e}_{\gamma }{e}_{\alpha }={e}_{\beta }=-{e}_{\alpha }{e}_{\gamma }.$

For each $x={\sum }_{0}^{7}\text{ }\text{ }{x}_{i}{e}_{i}\in O\left({x}_{i}\in R,i=0,1,\cdots ,7\right)$ , ${x}_{0}$ is called the scalar part of x and $\underset{_}{x}={\sum }_{1}^{7}\text{ }{x}_{i}{e}_{i}$ is termed its vector part. Then the norm of x is $|x|={\left({\sum }_{0}^{7}\text{ }{x}_{i}^{2}\right)}^{\frac{1}{2}}$ and its conjugate is defined by $\stackrel{¯}{x}={\sum }_{0}^{7}\text{ }\text{ }{x}_{i}{\stackrel{¯}{e}}_{i}={x}_{0}-\underset{_}{x}$ . We have $x\stackrel{¯}{x}=\stackrel{¯}{x}x={\sum }_{0}^{7}\text{ }{x}_{i}^{2}$ , $\stackrel{¯}{xy}=\stackrel{¯}{y}\stackrel{¯}{x}\left(x,y\in O\right)$ Hence, ${x}^{-1}=\frac{\stackrel{¯}{x}}{{|x|}^{2}}$ is the inverse of $x\left(\ne 0\right)$ .

Let $x={\sum }_{0}^{7}\text{ }\text{ }{x}_{i}{e}_{i},y={\sum }_{0}^{7}\text{ }\text{ }{y}_{i}{e}_{i}\in O\left({x}_{i},{y}_{i}\in R,i=0,1,\cdots ,7\right)$ , then

$xy={x}_{0}{y}_{0}-\underset{_}{x}\cdot \underset{_}{y}+{x}_{0}\underset{_}{y}+{y}_{0}\underset{_}{x}+\underset{_}{x}×\underset{_}{y},$ (2.1)

where $\underset{_}{x}\cdot \underset{_}{y}:={\sum }_{1}^{7}\text{ }{x}_{i}{y}_{i}$ is the inner product of vectors $\underset{_}{x},\underset{_}{y}$ and

$\begin{array}{c}\underset{_}{x}×\underset{_}{y}:={e}_{1}\left({A}_{23}+{A}_{45}-{A}_{67}\right)+{e}_{2}\left(-{A}_{13}+{A}_{46}+{A}_{57}\right)+{e}_{3}\left({A}_{12}+{A}_{47}-{A}_{56}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{e}_{4}\left(-{A}_{15}-{A}_{26}-{A}_{37}\right)+{e}_{5}\left({A}_{14}-{A}_{27}+{A}_{36}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{e}_{6}\left({A}_{17}+{A}_{24}-{A}_{35}\right)+{e}_{7}\left(-{A}_{16}+{A}_{25}+{A}_{34}\right)\end{array}$

is the cross product of vectors $\underset{_}{x},\underset{_}{y}$ , with

${A}_{ij}=\mathrm{det}\left(\begin{array}{cc}{x}_{i}& {x}_{j}\\ {y}_{i}& {y}_{j}\end{array}\right),\text{ }i,j=1,2,\cdots ,7.$

For any $x,y\in O$ , the inner product and cross product of their vector parts satisfy the following rules [8] :

$\left(\underset{_}{x}×\underset{_}{y}\right)\cdot \underset{_}{x}=0,\text{ }\left(\underset{_}{x}×\underset{_}{y}\right)\cdot \underset{_}{y}=0,\text{ }\underset{_}{x}||\underset{_}{y}⇔\underset{_}{x}×\underset{_}{y}=0,\text{ }\underset{_}{x}×\underset{_}{y}=-\underset{_}{y}×\underset{_}{x}.$

We usually utilize associator as an useful tool on ontonions since its non- associativity. Define the associator $\left[x,y,z\right]$ of any $x,y,z\in O$ by $\left[x,y,z\right]=\left(xy\right)z-x\left(yz\right)$ .

The octonions obey the following some weakened associative laws.

For any $x,y,z,u,v\in O$ , we have (see [7] )

$\left[x,y,z\right]=\left[y,z,x\right],\text{ }\left[x,z,y\right]=-\left[x,y,z\right],\text{ }\left[x,x,y\right]=0=\left[\stackrel{¯}{x},x,y\right]$ (2.2)

and the so-called Moufang identities [5]

$\left(uvu\right)x=u\left(v\left(ux\right)\right),\text{ }x\left(uvu\right)=\left(\left(xu\right)v\right)u,\text{ }u\left(xy\right)u=\left(ux\right)\left(uy\right).$

Proposition 2.1 ( [7] ). For any $i,j,k\in \left\{0,1,\cdots ,7\right\}$ , $\left[{e}_{i},{e}_{j},{e}_{k}\right]=0⇔ijk=0$ or $\left(i-j\right)\left(j-k\right)\left(k-i\right)=0$ or $\left({e}_{i}{e}_{j}\right){e}_{k}=±1$ .

Proposition 2.2 ( [7] ). Let ${e}_{i},{e}_{j},{e}_{k}$ be three different elements of $\left\{{e}_{1},{e}_{2},\cdots ,{e}_{7}\right\}$ and $\left({e}_{i}{e}_{j}\right){e}_{k}\ne ±1$ . Then $\left({e}_{i}{e}_{j}\right){e}_{k}=-{e}_{i}\left({e}_{j}{e}_{k}\right)$ .

Since octonions is an alternative algebra (see [3] [9] [10] ), we have the following power-associativity of octonions.

Proposition 2.3. Let ${x}_{1},{x}_{2},\cdots ,{x}_{k}\in O$ , $\left({l}_{1},\cdots ,{l}_{n}\right)$ be $n$ elements out of

$\left\{1,\cdots ,k\right\}$ repetitions being allowed and let ${\left({x}_{{l}_{1}}{x}_{{l}_{2}}\cdots {x}_{{l}_{n}}\right)}_{{\otimes }_{n}}$ be the product of $n$ octonions in a fixed associative order ${\otimes }_{n}$ . Then $\underset{\pi \left({l}_{1},\cdots ,{l}_{n}\right)}{\sum }{\left({x}_{{l}_{1}}{x}_{{l}_{2}}\cdots {x}_{{l}_{n}}\right)}_{{\otimes }_{n}}$ is

independent of the associative order ${\otimes }_{n}$ , where the sum runs over all distinguishable permutations of $\left({l}_{1},\cdots ,{l}_{n}\right)$

Proof. Let $x={\lambda }_{1}{x}_{1}+{\lambda }_{2}{x}_{2}+\cdots +{\lambda }_{k}{x}_{k}$ , then $\underset{\pi \left({l}_{1},\cdots ,{l}_{n}\right)}{\sum }{\left({x}_{{l}_{1}}{x}_{{l}_{2}}\cdots {x}_{{l}_{n}}\right)}_{{\otimes }_{n}}$ is just the coefficient of ${\lambda }_{{l}_{1}}{\lambda }_{{l}_{2}}\cdots {\lambda }_{{l}_{n}}$ in the product of ${x}^{n}=\underset{n\text{\hspace{0.17em}}x\text{s}}{\underset{︸}{\left(xx\cdots x\right)}}{\text{ }}_{{\otimes }_{n}}$ . By induction and (2.2), one can easily prove that ${x}^{n}=\underset{n\text{\hspace{0.17em}}x\text{s}}{\underset{︸}{\left(xx\cdots x\right)}}{\text{ }}_{{\otimes }_{n}}$ is independent of the associative order ${\otimes }_{n}$ for any $x\in O$ . Hence $\underset{\pi \left({l}_{1},\cdots ,{l}_{n}\right)}{\sum }{\left({x}_{{l}_{1}}{x}_{{l}_{2}}\cdots {x}_{{l}_{n}}\right)}_{{\otimes }_{n}}$ is also independent of the associative order ${\otimes }_{n}$ . ,

$\mu =\left({\mu }_{0},{\mu }_{1},\cdots ,{\mu }_{n}\right)$ is called a Stein-Weiss conjugate harmonic system if they satisfy the following equations (see [11] ):

$\underset{i=0}{\overset{n}{\sum }}\frac{\partial {\mu }_{i}}{\partial {x}_{i}}=0,\text{ }\frac{\partial {\mu }_{i}}{\partial {x}_{j}}=\frac{\partial {\mu }_{j}}{\partial {x}_{i}}\text{ }\left(0\le i

It is easy to see that if $F\left({x}_{0},{x}_{1},\cdots ,{x}_{7}\right)=\left({f}_{0},{f}_{1},\cdots ,{f}_{7}\right)$ is a Stein-Weiss conjugate harmonic system in an open set $\Omega$ of ${R}^{8}$ , then there exists a real- valued harmonic function $\Phi$ in $\Omega$ such that F is the gradient of $\Phi$ . Thus $\stackrel{¯}{F}={f}_{0}{e}_{0}-{f}_{1}{e}_{1}-\cdots -{f}_{7}{e}_{7}=\stackrel{¯}{D}\Phi$ is an $O$ -analytic function. But inversely, this is not true [12] .

Example. Observe the $O$ -analytic function $g\left(x\right)=\left({x}_{6}^{2}-{x}_{7}^{2}\right){e}_{2}-2{x}_{6}{x}_{7}{e}_{3}$ . Since

$\frac{\partial {g}_{2}}{\partial {x}_{6}}=2{x}_{6}\ne 0=\frac{\partial {g}_{6}}{\partial {x}_{2}},$

$\stackrel{¯}{g}$ 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 $f\left(x\right)$ is a left $O$ -analytic function in $\Omega$ which containing the origin, then it can be developed into Taylor series

$f\left(x\right)=\underset{k=0}{\overset{\infty }{\sum }}\underset{\left({l}_{1},\cdots ,{l}_{k}\right)}{\sum }{V}_{{l}_{1}\cdots {l}_{k}}\left(x\right){\partial }_{{x}_{{l}_{1}}}\cdots {\partial }_{{x}_{{l}_{k}}}f\left(0\right),$

and if $f\left(x\right)$ is a right $O$ -analytic function, then the Taylor series of $f$ at the origin is given by

$f\left(x\right)=\underset{k=0}{\overset{\infty }{\sum }}\underset{\left({l}_{1},\cdots ,{l}_{k}\right)}{\sum }{\partial }_{{x}_{{l}_{1}}}\cdots {\partial }_{{x}_{{l}_{k}}}f\left(0\right){V}_{{l}_{1}\cdots {l}_{k}}\left(x\right),$

where $\left({l}_{1},\cdots ,{l}_{k}\right)$ runs over all possible combinations of $k$ elements out of $\left\{1,\cdots ,7\right\}$ repetitions being allowed.

The polynomials ${V}_{{l}_{1}\cdots {l}_{k}}$ of order $k$ in Theorem A is defined by

${V}_{{l}_{1}\cdots {l}_{k}}\left(x\right)=\frac{1}{k!}\underset{\pi \left({l}_{1},\cdots ,{l}_{k}\right)}{\sum }\left(\cdots \left(\left({z}_{{l}_{1}}{z}_{{l}_{2}}\right){z}_{{l}_{3}}\right)\cdots \right){z}_{{l}_{k}},$

where the sum runs over all distinguishable permutations of $\left({l}_{1},\cdots ,{l}_{k}\right)$ and ${z}_{{l}_{j}}={x}_{{l}_{j}}{e}_{0}-{x}_{0}{e}_{{l}_{j}},j=1,\cdots ,k$ .

We have the following uniqueness theorem for $O$ -analytic functions [7] .

Proposition 2.4. If $f$ is left (right) $O$ -analytic in an open connect set $\Omega \subset {R}^{8}$ and vanishes in the open set $E\subset \Omega \cap \left\{{x}_{0}={a}_{0}\right\}\ne \varnothing$ , then $f$ is identically zero in $\Omega$ .

Proof. Without loss of generality, we let $E$ which containing the origin and let ${x}_{0}=0$ . Then $f$ can be developed into Taylor series

$f\left(x\right)=\underset{k=0}{\overset{\infty }{\sum }}\underset{\left({l}_{1},\cdots ,{l}_{k}\right)}{\sum }{V}_{{l}_{1}\cdots {l}_{k}}\left(x\right){\partial }_{{x}_{{l}_{1}}}\cdots {\partial }_{{x}_{{l}_{k}}}f\left(0\right).$

Thus we have

$f\left(\underset{_}{x}\right)=\underset{k=0}{\overset{\infty }{\sum }}\underset{\left({l}_{1},\cdots ,{l}_{k}\right)}{\sum }{x}_{{l}_{1}}{x}_{{l}_{2}}\cdots {x}_{{l}_{k}}{\partial }_{{x}_{{l}_{1}}}\cdots {\partial }_{{x}_{{l}_{k}}}f\left(0\right)\equiv 0.$

By the uniqueness of the Taylor series for the real analytic function, we have ${\partial }_{{x}_{{l}_{1}}}\cdots {\partial }_{{x}_{{l}_{k}}}f\left(0\right)=0$ for any $\left({l}_{1},\cdots ,{l}_{k}\right)\in {\left\{1,2,\cdots ,7\right\}}^{7}$ and $k\in N$ . This shows that $f$ is identically zero in $E$ and also in $\Omega$ . ,

For more references about octonions and octonionic analysis, we refer the reader to [7] [13] - [20] .

3. Sufficient Conditions

In what follows we consider the complexification of $O$ , it is denoted by ${O}^{c}$ .

Thus, $ℤ\in {O}^{c}$ is of the form $ℤ={\sum }_{0}^{7}\text{ }{ℤ}_{i}{e}_{i},{ℤ}_{i}\in C$ . ${ℤ}_{0}$ and $\underset{_}{ℤ}={\sum }_{0}^{7}\text{ }{ℤ}_{i}{e}_{i}$ are still

called the scalar part and vector part, respectively. The norm of $ℤ\in {O}^{c}$ is

$|ℤ|={\left({\sum }_{0}^{7}{|{ℤ}_{i}|}^{2}\right)}^{\frac{1}{2}}$ and its conjugate is defined by $\stackrel{¯}{ℤ}={\sum }_{0}^{7}\text{ }\text{ }{\stackrel{¯}{ℤ}}_{i}{\stackrel{¯}{e}}_{i}$ , where ${\stackrel{¯}{ℤ}}_{i}$ is of the

conjugate in the complex numbers. We can easily show that for any $ℤ,{ℤ}^{\prime }\in {O}^{c}$ , $|ℤ{ℤ}^{\prime }|\le \sqrt{2}|ℤ||{ℤ}^{\prime }|$ . For any $ℤ\in {O}^{c}$ , we may rewrite $ℤ$ as $ℤ=x+iy$ , where $x,y\in O$ . The multiplication rules in ${O}^{c}$ is the same as in (2.1). Note that ${O}^{c}$ is no longer a division algebra. Finally, the properties of associator in (2.2) except that $\left[ℤ,\stackrel{¯}{ℤ},\mathbb{U}\right]=0$ are also true for any $ℤ,\mathbb{U},\mathbb{V}\in {O}^{c}$ :

$\left[ℤ,\mathbb{U},\mathbb{V}\right]=\left[\mathbb{U},\mathbb{V},ℤ\right],\text{ }\left[ℤ,\mathbb{V},\mathbb{U}\right]=-\left[ℤ,\mathbb{U},\mathbb{V}\right],\text{ }\left[ℤ,ℤ,\mathbb{U}\right]=0.$ (3.1)

Example. Let $ℤ={e}_{1}+i{e}_{2},\mathbb{U}={e}_{4}$ , then

$\left[ℤ,\stackrel{¯}{ℤ},\mathbb{U}\right]=\left[{e}_{1}+i{e}_{2},-{e}_{1}+i{e}_{2},{e}_{4}\right]=i\left[{e}_{1},{e}_{2},{e}_{4}\right]-i\left[{e}_{2},{e}_{1},{e}_{4}\right]=4i{e}_{7}\ne 0.$

By (3.1) we can get the following lemma, which is useful to deduce our results.

Lemma 3.1. Let $ℤ,\mathbb{U},\mathbb{V}\in {O}^{c}$ and there exists complex numbers $\lambda$ and $\mu \left(|\lambda |+|\mu |\ne 0\right)$ such that $\lambda \underset{_}{ℤ}+\mu \underset{_}{\mathbb{U}}=0$ or $\lambda \underset{_}{\mathbb{U}}+\mu \underset{_}{\mathbb{V}}=0$ or $\lambda \underset{_}{\mathbb{V}}+\mu \underset{_}{ℤ}=0$ , then $\left[ℤ,\mathbb{U},\mathbb{V}\right]=0$ .

For functions, f, under study will be defined in an open set $\Omega$ of ${R}^{8}$ and

take values in ${O}^{c}$ , with the form $f\left(x\right)={\sum }_{0}^{7}\text{ }{f}_{i}\left(x\right){e}_{i}$ , where ${f}_{i}\left(x\right)\left(i=0,1,\cdots ,7\right)$

are the complex-valued functions.

Hence, we say that, a function $f\left(x\right)=g\left(x\right)+ih\left(x\right)$ is left ${O}^{c}$ -analytic in an open set $\Omega$ of ${R}^{8}$ , if $g\left(x\right)$ and $h\left(x\right)$ are the left $O$ -analytic functions, since

$Df=0⇔Dg=Dh=0,$

where $D={\sum }_{i=0}^{7}\frac{\partial }{\partial {x}_{i}}{e}_{i}$ is the Dirac operator as in Section 1.

In the case of ${O}^{c}$ , we call $f\left(x\right)=g\left(x\right)+ih\left(x\right)$ a complex Stein-Weiss conjugate harmonic system, if $g\left(x\right),h\left(x\right)$ are the Stein-Weiss conjugate harmonic systems. A left (right) ${O}^{c}$ -analytic functions $g\left(x\right)$ also have the Taylor expansion as in Theorem A.

Now we consider the product $f\left(x\right)g\left(x\right)$ of two left ${O}^{c}$ -analytic functions $f\left(x\right),g\left(x\right)$ in $\Omega$ . In general, $f\left(x\right)g\left(x\right)$ is no longer left ${O}^{c}$ -analytic in $\Omega$ . But, in some particular cases, the product $f\left(x\right)g\left(x\right)$ can maintain the analyticity for two left ${O}^{c}$ -analytic functions $f\left(x\right)$ and $g\left(x\right)$ .

Theorem 3.2. Let $f\left(x\right),g\left(x\right)$ be two left ${O}^{c}$ -analytic functions in $\Omega$ . Then $f\left(x\right)g\left(x\right)$ is also left $O$ -analytic in $\Omega$ if $f\left(x\right),g\left(x\right)$ satisfy one of the following conditions:

1) $f\left(x\right)$ or $g\left(x\right)$ is a complex constant function.

2) $\stackrel{¯}{f\left(x\right)}$ is a complex Stein-Weiss conjugate harmonic system in $\Omega$ and $g\left(x\right)$ is an ${O}^{c}$ -constant function.

3) $f\left(x\right)$ is of the form $f\left(x\right)={f}_{0}{e}_{0}+{f}_{i}{e}_{i}\text{\hspace{0.17em}}\left(i\in \left\{1,2,\cdots ,7\right\}\right)$ and $f\left(x\right),g\left(x\right)$ depend only on ${x}_{0}$ and ${x}_{i}$ , where ${f}_{0},{f}_{i}$ are the complex-valued functions.

4) $f\left(x\right)$ and $g\left(x\right)$ belong to the following class

$\mathfrak{S}=\left\{h\left(x\right)|Dh\left(x\right)=0,\underset{_}{h\left(x\right)}=\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{h}_{1}\left(x\right){e}_{i},{h}_{1}\left(x\right)\in {C}^{1}\left(\Omega ,C\right)\right\}.$ (3.2)

5) $f\left(x\right)$ is of the form $f\left(x\right)={f}_{0}{e}_{0}+{f}_{\alpha }{e}_{\alpha }+{f}_{\beta }{e}_{\beta }+{f}_{\gamma }{e}_{\gamma }$ , $g={c}_{0}{e}_{0}+{c}_{\alpha }{e}_{\alpha }+{c}_{\beta }{e}_{\beta }+{c}_{\gamma }{e}_{\gamma }$ is a constant function, where $\left(\alpha ,\beta ,\gamma \right)\in W$ , ${c}_{0},{c}_{\alpha },{c}_{\beta },{c}_{\gamma }\in C$ and $f\left(x\right)$ depends only on ${x}_{0},{x}_{\alpha },{x}_{\beta },{x}_{\gamma }$ .

Proof. 1) The proof is trivial.

2) In view of Proposition 2.1 we have $\left[{e}_{i},{e}_{j},\lambda \right]=0$ when $i=0$ or $j=0$ or $i=j$ for any $\lambda \in {O}^{c}$ . Then we have

$\begin{array}{c}D\left(f\lambda \right)=\underset{i,j=0}{\overset{7}{\sum }}\frac{\partial {f}_{j}}{\partial {x}_{i}}{e}_{i}\left({e}_{j}\lambda \right)\\ =\underset{i,j=0}{\overset{7}{\sum }}\frac{\partial {f}_{j}}{\partial {x}_{i}}\left({e}_{i}{e}_{j}\right)\lambda -\underset{i,j=0}{\overset{7}{\sum }}\frac{\partial {f}_{j}}{\partial {x}_{i}}\left[{e}_{i},{e}_{j},\lambda \right]\\ =\left(Df\right)\lambda -\underset{i,j=0}{\overset{7}{\sum }}\frac{\partial {f}_{j}}{\partial {x}_{i}}\left[{e}_{i},{e}_{j},\lambda \right]\\ =\left(Df\right)\lambda -\underset{1\le i\ne j\le 7}{\sum }\frac{\partial {f}_{j}}{\partial {x}_{i}}\left[{e}_{i},{e}_{j},\lambda \right].\end{array}$

Since $\stackrel{¯}{f}$ is a complex Stein-Weiss conjugate harmonic system, thus $Df=0$

and $\frac{\partial {f}_{j}}{\partial {x}_{i}}=\frac{\partial {f}_{i}}{\partial {x}_{j}}$ for $i,j\ge 1,i\ne j$ . But $\left[{e}_{j},{e}_{i},\lambda \right]=-\left[{e}_{i},{e}_{j},\lambda \right]$ , therefore

$D\left(f\lambda \right)=-\underset{1\le i\ne j\le 7}{\sum }\frac{\partial {f}_{j}}{\partial {x}_{i}}\left[{e}_{i},{e}_{j},\lambda \right]=\underset{1\le i

3) Since $f\left(x\right),g\left(x\right)$ are only related to variables ${x}_{0}$ and ${x}_{i}$ , we have

$\begin{array}{c}D\left(fg\right)=\left(\frac{\partial }{\partial {x}_{0}}+\frac{\partial }{\partial {x}_{i}}{e}_{i}\right)\left(\left({f}_{0}+{f}_{i}{e}_{i}\right)g\right)\\ =\frac{\partial f}{\partial {x}_{0}}g+{e}_{i}\left(\left(\frac{\partial {f}_{0}}{\partial {x}_{i}}+\frac{\partial {f}_{i}}{\partial {x}_{i}}{e}_{i}\right)g\right)+f\frac{\partial g}{\partial {x}_{0}}+{e}_{i}\left(\left({f}_{0}+{f}_{i}{e}_{i}\right)\frac{\partial g}{\partial {x}_{i}}\right).\end{array}$

By Lemma 3.1 it follows that

${e}_{i}\left(\left(\frac{\partial {f}_{0}}{\partial {x}_{i}}+\frac{\partial {f}_{i}}{\partial {x}_{i}}{e}_{i}\right)g\right)=\left({e}_{i}\left(\frac{\partial {f}_{0}}{\partial {x}_{i}}+\frac{\partial {f}_{i}}{\partial {x}_{i}}{e}_{i}\right)\right)g=\left({e}_{i}\frac{\partial f}{\partial {x}_{i}}\right)g$

and

${e}_{i}\left(\left({f}_{0}+{f}_{i}{e}_{i}\right)\frac{\partial g}{\partial {x}_{i}}\right)=\left({e}_{i}\left({f}_{0}+{f}_{i}{e}_{i}\right)\right)\frac{\partial g}{\partial {x}_{i}}=\left(\left({f}_{0}+{f}_{i}{e}_{i}\right){e}_{i}\right)\frac{\partial g}{\partial {x}_{i}}=\left({f}_{0}+{f}_{i}{e}_{i}\right)\left({e}_{i}\frac{\partial g}{\partial {x}_{i}}\right).$

Thus we get

$D\left(fg\right)=\frac{\partial f}{\partial {x}_{0}}g+\left({e}_{i}\frac{\partial f}{\partial {x}_{i}}\right)g+f\frac{\partial g}{\partial {x}_{0}}+f\left({e}_{i}\frac{\partial g}{\partial {x}_{i}}\right)=\left(Df\right)g+f\left(Dg\right)=0.$

4) Let $f\left(x\right)={f}_{0}{e}_{0}+{\sum }_{i=1}^{7}\text{ }{f}_{1}{e}_{i}$ and $g\left(x\right)={g}_{0}{e}_{0}+{\sum }_{i=1}^{7}\text{ }{g}_{1}{e}_{i}$ , then we have

$\begin{array}{c}D\left(f\left(x\right)g\left(x\right)\right)=\underset{j=0}{\overset{7}{\sum }}\text{ }\text{ }{e}_{j}\frac{\partial }{\partial {x}_{j}}\left(\left({f}_{0}{e}_{0}+{f}_{1}\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\right)\left({g}_{0}{e}_{0}+{g}_{1}\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\right)\right)\\ =\underset{j=0}{\overset{7}{\sum }}\text{ }\text{ }{e}_{j}\left(\left(\frac{\partial {f}_{0}}{\partial {x}_{j}}{e}_{0}+\frac{\partial {f}_{1}}{\partial {x}_{j}}\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\right)\left({g}_{0}{e}_{0}+{g}_{1}\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\right)\right)\\ \text{\hspace{0.17em}}+\text{\hspace{0.17em}}\underset{j=0}{\overset{7}{\sum }}\text{ }\text{ }{e}_{j}\left(\left({f}_{0}{e}_{0}+{f}_{1}\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\right)\left(\frac{\partial {g}_{0}}{\partial {x}_{j}}{e}_{0}+\frac{\partial {g}_{1}}{\partial {x}_{j}}\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\right)\right).\end{array}$

By Lemma 3.1 we get

${e}_{j}\left(\left(\frac{\partial {f}_{0}}{\partial {x}_{j}}{e}_{0}+\frac{\partial {f}_{1}}{\partial {x}_{j}}\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\right)\left({g}_{0}{e}_{0}+{g}_{1}\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\right)\right)=\left({e}_{j}\frac{\partial f}{\partial {x}_{j}}\right)g$

and

$\begin{array}{l}{e}_{j}\left(\left({f}_{0}{e}_{0}+{f}_{1}\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\right)\left(\frac{\partial {g}_{0}}{\partial {x}_{j}}{e}_{0}+\frac{\partial {g}_{1}}{\partial {x}_{j}}\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\right)\right)\\ ={e}_{j}\left(\left(\frac{\partial {g}_{0}}{\partial {x}_{j}}{e}_{0}+\frac{\partial {g}_{1}}{\partial {x}_{j}}\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\right)\left({f}_{0}{e}_{0}+{f}_{1}\underset{i=1}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\right)\right)\\ =\left({e}_{j}\frac{\partial g}{\partial {x}_{j}}\right)f.\end{array}$

Hence we obtain

$D\left(f\left(x\right)g\left(x\right)\right)=\underset{j=0}{\overset{7}{\sum }}\left(\left({e}_{j}\frac{\partial f}{\partial {x}_{j}}\right)g+\left({e}_{j}\frac{\partial g}{\partial {x}_{j}}\right)f\right)=\left(Df\right)g+\left(Dg\right)f=0.$

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 $f\left(x\right),g\left(x\right)\in \mathfrak{S}$ , then $f\left(x\right)g\left(x\right)\in \mathfrak{S}$ ; that is, the multiply operation in $\mathfrak{S}$ is closed. Also, the division operation is closed in $\mathfrak{S}$ .

Actually, let $f\left(x\right)={f}_{0}\left(x\right)+{\sum }_{i=1}^{7}\text{ }{f}_{1}\left(x\right){e}_{i}\in \mathfrak{S}$ , assume ${f}_{0}^{2}+7{f}_{1}^{2}\ne 0$ , then

${\left(f\left(x\right)\right)}^{-1}=\frac{{f}_{0}-{f}_{1}\left({e}_{1}+{e}_{2}+\cdots +{e}_{7}\right)}{{f}_{0}^{2}+7{f}_{1}^{2}}.$

Thus we have

$\begin{array}{l}D{\left(f\left(x\right)\right)}^{-1}=\underset{i=0}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\frac{\partial {\left(f\left(x\right)\right)}^{-1}}{\partial {x}_{i}}\\ =\underset{i=0}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\left({\left({f}_{0}^{2}+7{f}_{1}^{2}\right)}^{-1}\left(\frac{\partial {f}_{0}}{\partial {x}_{i}}-\frac{\partial {f}_{1}}{\partial {x}_{i}}\left({e}_{1}+{e}_{2}+\cdots +{e}_{7}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-\left({f}_{0}-{f}_{1}\left({e}_{1}+{e}_{2}+\cdots +{e}_{7}\right)\right)\left(2{f}_{0}\frac{\partial {f}_{0}}{\partial {x}_{i}}+14{f}_{1}\frac{\partial {f}_{1}}{\partial {x}_{i}}\right){\left({f}_{0}^{2}+7{f}_{1}^{2}\right)}^{-2}\right)\\ =\underset{i=0}{\overset{7}{\sum }}\text{ }\text{ }{e}_{i}\left(\left(\frac{\partial {f}_{0}}{\partial {x}_{i}}+\frac{\partial {f}_{1}}{\partial {x}_{i}}\left({e}_{1}+\cdots +{e}_{7}\right)\right)\left(7{f}_{1}^{2}-{f}_{0}^{2}+2{f}_{0}{f}_{1}\left({e}_{1}+\cdots +{e}_{7}\right)\right){\left({f}_{0}^{2}+7{f}_{1}^{2}\right)}^{-2}\right)\end{array}$

$\begin{array}{l}=\underset{i=0}{\overset{7}{\sum }}\left({e}_{i}\left(\frac{\partial {f}_{0}}{\partial {x}_{i}}+\frac{\partial {f}_{1}}{\partial {x}_{i}}\left({e}_{1}+\cdots +{e}_{7}\right)\right)\right)\left(7{f}_{1}^{2}-{f}_{0}^{2}+2{f}_{0}{f}_{1}\left({e}_{1}+\cdots +{e}_{7}\right)\right){\left({f}_{0}^{2}+7{f}_{1}^{2}\right)}^{-2}\\ =\left(Df\left(x\right)\right)\left(7{f}_{1}^{2}-{f}_{0}^{2}+2{f}_{0}{f}_{1}\left({e}_{1}+\cdots +{e}_{7}\right)\right){\left({f}_{0}^{2}+7{f}_{1}^{2}\right)}^{-2}\\ =0.\end{array}$

An element belongs to $\mathfrak{S}$ is the exponential function:

$\mathrm{exp}\left(x\right)={\text{e}}^{{x}_{1}+\cdots +{x}_{7}}\left(\mathrm{cos}\left({x}_{0}\sqrt{7}\right){e}_{0}+\left(-\frac{1}{\sqrt{7}}\left({e}_{1}+\cdots +{e}_{7}\right)\right)\mathrm{sin}\left({x}_{0}\sqrt{7}\right)\right).$ (3.3)

The results in Theorem 3.2 also hold on octonions(no complexification), since ${O}^{c}$ contains $O$ . If one switch the locations of $f\left(x\right),g\left(x\right)$ , 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 ${O}^{c}$ -analytic function and an ${O}^{c}$ -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 $f\left(x\right)$ is a left ${O}^{c}$ -analytic function and $\lambda$ is a complex constant, or $\stackrel{¯}{f\left(x\right)}$ is a complex Stein-Weiss conjugate harmonic system and $\lambda$ is an ${O}^{c}$ -constant, then $f\left(x\right)\lambda$ is a left ${O}^{c}$ - analytic function. In what follows we will see that these conditions are also necessary in some sense.

Theorem 4.1. Let $\lambda \in {O}^{c}$ , then $f\lambda$ is a left ${O}^{c}$ -analytic function for any left ${O}^{c}$ -analytic functions $f$ if and only if $\lambda \in C$ .

Proof. We only prove the necessity. Taking a left ${O}^{c}$ -analytic function $f={x}_{1}{e}_{2}-{x}_{0}{e}_{3}$ , then

$\begin{array}{c}D\left(f\lambda \right)=-\underset{i,j,k=0}{\overset{7}{\sum }}\frac{\partial {f}_{j}}{\partial {x}_{i}}{\lambda }_{k}\left[{e}_{i},{e}_{j},{e}_{k}\right]=\underset{k=1}{\overset{7}{\sum }}\frac{\partial {f}_{2}}{\partial {x}_{1}}{\lambda }_{k}\left[{e}_{2},{e}_{1},{e}_{k}\right]=\underset{k=4}{\overset{7}{\sum }}\text{ }\text{ }{\lambda }_{k}\left[{e}_{2},{e}_{1},{e}_{k}\right]\\ ={\lambda }_{4}\left[{e}_{2},{e}_{1},{e}_{4}\right]+{\lambda }_{5}\left[{e}_{2},{e}_{1},{e}_{5}\right]+{\lambda }_{6}\left[{e}_{2},{e}_{1},{e}_{6}\right]+{\lambda }_{7}\left[{e}_{2},{e}_{1},{e}_{7}\right]\\ =-2{\lambda }_{4}{e}_{7}+2{\lambda }_{5}{e}_{6}-2{\lambda }_{6}{e}_{5}+2{\lambda }_{7}{e}_{4}.\end{array}$

Thus ${\lambda }_{4}={\lambda }_{5}={\lambda }_{6}={\lambda }_{7}=0$ . A similar technique yields ${\lambda }_{1}={\lambda }_{2}={\lambda }_{3}=0$ . Hence $\lambda \in C$ . ,

Theorem 4.2. Let $f\in {C}^{1}\left(\Omega ,{O}^{c}\right)$ . Then $D\left(f\lambda \right)=0$ for any $\lambda \in {O}^{c}$ if and only if $f$ is a complex Stein-Weiss conjugate harmonic system in $\Omega$ .

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 ${O}^{c}$ -valued function $f$ , such that the equality $\left[\lambda ,f\left(x\right),D\right]=0$ holds for any constant $\lambda \in {O}^{c}$ .

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 $\left[\lambda ,f\left(x\right),D\right]=0$ for $f\left(x\right)$ 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 $f\in {C}^{1}\left(\Omega ,{O}^{c}\right)$ . Then $\left[D,f,\lambda \right]=0\text{\hspace{0.17em}}\left(\left[\lambda ,f,D\right]=0\right)$ for any $\lambda \in {O}^{c}$ if and only if

$\frac{\partial {f}_{i}}{\partial {x}_{j}}=\frac{\partial {f}_{j}}{\partial {x}_{i}},\text{ }i,j=1,2,\cdots ,7.$ (4.1)

Proof. By Proposition 2.1, we have

$\left[D,f,\lambda \right]=\underset{i,j=0}{\overset{7}{\sum }}\frac{\partial {f}_{j}}{\partial {x}_{i}}\left[{e}_{i},{e}_{j},\lambda \right]=\underset{1\le i

If $f$ satisfies (4.1), then $\left[D,f,\lambda \right]=0$ .

Inversely, let $\left(\alpha ,\beta ,\gamma \right)\in W$ , $\left\{1,2,\cdots ,7\right\}\\left\{\alpha ,\beta ,\gamma \right\}=\left\{{t}_{1},{t}_{2},{t}_{3},{t}_{4}\right\}$ and

${e}_{{t}_{1}}{e}_{{t}_{2}}={e}_{\gamma }=-{e}_{{t}_{2}}{e}_{{t}_{1}},\text{\hspace{0.17em}}{e}_{{t}_{3}}{e}_{{t}_{4}}={e}_{\gamma }=-{e}_{{t}_{4}}{e}_{{t}_{3}}.$

From Propositions 2.1 and 2.2 we have $\left[{e}_{\alpha },{e}_{\beta },{e}_{t}\right]=0$ and $\left[{e}_{\alpha },{e}_{\beta },{e}_{t}\right]=2\left({e}_{\alpha }{e}_{\beta }\right){e}_{t}=2{e}_{\gamma }{e}_{t}$ when $t=\alpha ,\beta ,\gamma$ and $t={t}_{1},{t}_{2},{t}_{3},{t}_{4}$ , respectively. Hence, taking $\lambda ={e}_{{t}_{1}}$ it follows that

$\begin{array}{l}\left[D,f,{e}_{{t}_{1}}\right]\\ =\underset{1\le i (4.2)

Similarly, we take $\lambda ={e}_{{t}_{3}}$ , then

$\left[D,f,{e}_{{t}_{3}}\right]=2\left(\frac{\partial {f}_{\beta }}{\partial {x}_{\alpha }}-\frac{\partial {f}_{\alpha }}{\partial {x}_{\beta }}+\frac{\partial {f}_{{t}_{2}}}{\partial {x}_{{t}_{1}}}-\frac{\partial {f}_{{t}_{1}}}{\partial {x}_{{t}_{2}}}\right){e}_{{t}_{4}}+\underset{s\ne {t}_{4}}{\sum }{h}_{s}{e}_{s},$ (4.3)

Also we can get

$\left[D,f,{e}_{\alpha }\right]=2\left(\frac{\partial {f}_{{t}_{2}}}{\partial {x}_{{t}_{1}}}-\frac{\partial {f}_{{t}_{1}}}{\partial {x}_{{t}_{2}}}+\frac{\partial {f}_{{t}_{4}}}{\partial {x}_{{t}_{3}}}-\frac{\partial {f}_{{t}_{3}}}{\partial {x}_{{t}_{4}}}\right){e}_{\beta }+\underset{s\ne \beta }{\sum }\text{ }{y}_{s}{e}_{s}.$ (4.4)

If we require $\left[D,f,\lambda \right]=0$ for any constants $\lambda \in {O}^{c}$ , from (4.2), (4.3) and (4.4) we obtain

$\left\{\begin{array}{l}\frac{\partial {f}_{\beta }}{\partial {x}_{\alpha }}-\frac{\partial {f}_{\alpha }}{\partial {x}_{\beta }}+\frac{\partial {f}_{{t}_{4}}}{\partial {x}_{{t}_{3}}}-\frac{\partial {f}_{{t}_{3}}}{\partial {x}_{{t}_{4}}}=0,\\ \frac{\partial {f}_{\beta }}{\partial {x}_{\alpha }}-\frac{\partial {f}_{\alpha }}{\partial {x}_{\beta }}+\frac{\partial {f}_{{t}_{2}}}{\partial {x}_{{t}_{1}}}-\frac{\partial {f}_{{t}_{1}}}{\partial {x}_{{t}_{2}}}=0,\\ \frac{\partial {f}_{{t}_{2}}}{\partial {x}_{{t}_{1}}}-\frac{\partial {f}_{{t}_{1}}}{\partial {x}_{{t}_{2}}}+\frac{\partial {f}_{{t}_{4}}}{\partial {x}_{{t}_{3}}}-\frac{\partial {f}_{{t}_{3}}}{\partial {x}_{{t}_{4}}}=0.\end{array}$

Combining above three equations with the randomicity of $\left(\alpha ,\beta ,\gamma \right)$ we have (4.1) holds. ,

Proof of Theorem 4.2. The sufficient from Theorem 3.2(b). Inversely, if we take $\lambda =1$ in $D\left(f\lambda \right)=0$ it follows that $f$ is a left ${O}^{c}$ -analytic function. Thus for any $\lambda \in {O}^{c}$ , we have

$D\left(f\lambda \right)=\left(Df\right)\lambda -\left[D,f,\lambda \right]=-\left[D,f,\lambda \right]=0.$

By Theorem 4.3 we get that $f$ satisfies (4.1). On the other hand,

$Df=\left(\frac{\partial }{\partial {x}_{0}}+\nabla \right)\left({f}_{0}+\underset{_}{f}\right)=\frac{\partial {f}_{0}}{\partial {x}_{0}}-\nabla \cdot \underset{_}{f}+\frac{\partial \underset{_}{f}}{\partial {x}_{0}}+\nabla {f}_{0}+\nabla ×\underset{_}{f}=0.$ (4.5)

From (4.1) it easily to get $\nabla ×\underset{_}{f}=0$ , again by (4.5) it follows that

$\frac{\partial {f}_{0}}{\partial {x}_{0}}-\nabla \cdot \underset{_}{f}+\frac{\partial \underset{_}{f}}{\partial {x}_{0}}+\nabla {f}_{0}=0,$

namely

$\frac{\partial {f}_{0}}{\partial {x}_{0}}-\nabla \cdot \underset{_}{f}=0,\text{ }\frac{\partial \underset{_}{f}}{\partial {x}_{0}}+\nabla {f}_{0}=0.$

Combining this with (4.1) it shows that $\stackrel{¯}{f}$ is a complex Stein-Weiss conjugate harmonic system in $\Omega$ . ,

5. Some Applications and Relations with the C-K Products

From Theorem A we can see that ${V}_{{l}_{1}\cdots {l}_{k}}\left(x\right)$ are the basic components for (left) $O$ -analytic functions. It is proved in [13] that the polynomials ${V}_{{l}_{1}\cdots {l}_{k}}\left(x\right)$ are all $O$ -analytic functions, therefore they are the suitable substitutions of the polynomial ${z}^{k}$ in $C$ .

Again from Theorem A, since ${V}_{{l}_{1}\cdots {l}_{k}}\left(x\right){\lambda }_{{l}_{1}\cdots {l}_{k}}$ is an item in the Taylor expansion of a left $O$ -analytic function, ${V}_{{l}_{1}\cdots {l}_{k}}\left(x\right){\lambda }_{{l}_{1}\cdots {l}_{k}}$ should be also a left $O$ analytic function. Applying Theorem 4.2, the conjugate of ${V}_{{l}_{1}\cdots {l}_{k}}\left(x\right)$ is probably a Stein-Weiss conjugate harmonic system. The following theorem prove this is true.

Theorem 5.1. For any combination $\left({l}_{1},\cdots ,{l}_{k}\right)$ of $k$ elements out of $\left\{1,\cdots ,7\right\}$ repetitions being allowed, ${\stackrel{¯}{V}}_{{l}_{1}\cdots {l}_{k}}\left(x\right)$ is a Stein-Weiss conjugate harmonic system in ${R}^{8}$ .

Proof. Let ${s}_{i}\left(i=1,\cdots ,7\right)$ be the appearing times of $i$ in $\left({l}_{1},\cdots ,{l}_{k}\right)$ . Hence the following equality

${V}_{{l}_{1}\cdots {l}_{k}}\left(x\right)=\stackrel{¯}{D}{\Phi }_{{s}_{1}\cdots {s}_{7}}\left(x\right)$ (5.1)

shows that ${\stackrel{¯}{V}}_{{l}_{1}\cdots {l}_{k}}\left(x\right)$ is a Stein-Weiss conjugate harmonic system in ${R}^{8}$ , where

${\Phi }_{{s}_{1}\cdots {s}_{7}}\left(x\right)=\underset{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\kappa }_{i}=0\\ i=1,\cdots ,7\end{array}}{\overset{\left[\frac{{s}_{i}}{2}\right]}{\sum }}\left\{\frac{{\left(-1\right)}^{\kappa }\kappa !{x}_{0}^{2\kappa +1}}{\left(2\kappa +1\right)!}\underset{j=1}{\overset{7}{\prod }}\frac{{x}_{j}^{{s}_{j}-2{\kappa }_{j}}}{{\kappa }_{j}!\left({s}_{j}-2{\kappa }_{j}\right)!}\right\}$

is a real-valued harmonic function of order $\left({s}_{1}+{s}_{2}+\cdots +{s}_{7}+1\right)$ with $\kappa ={\sum }_{i=1}^{7}\text{ }\text{ }{\kappa }_{i}$ .

Actually, put ${x}_{0}=0$ , the both sides of (5.1) equal to $\frac{1}{{s}_{1}!{s}_{2}!\cdots {s}_{7}!}{x}_{1}^{{s}_{1}}\cdots {x}_{7}^{{s}_{7}}$ . On

the other hand, ${V}_{{l}_{1}\cdots {l}_{k}}\left(x\right)$ is left $O$ -analytic in ${R}^{8}$ . 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 ${V}_{{l}_{1}\cdots {l}_{k}}\left(x\right){\lambda }_{{l}_{1}\cdots {l}_{k}}$ are left ${O}^{c}$ -analytic functions for any ${\lambda }_{{l}_{1}\cdots {l}_{k}}\in {O}^{c}$ . Hence the following series

$\underset{k=0}{\overset{\infty }{\sum }}\underset{\left({l}_{1},\cdots ,{l}_{k}\right)}{\sum }{V}_{{l}_{1}\cdots {l}_{k}}\left(x\right){\lambda }_{{l}_{1}\cdots {l}_{k}}$ (5.2)

is a left ${O}^{c}$ -analytic function in some open neighborhood $\Lambda$ of the origin if $\left\{{\lambda }_{{l}_{1}\cdots {l}_{k}}\right\}$ satisfies certain bounded conditions.

Theorem 5.2. For any combination $\left({l}_{1},\cdots ,{l}_{k}\right)$ of k elements out of $\left\{1,\cdots ,7\right\}$

repetitions being allowed, let ${\lambda }_{{l}_{1}\cdots {l}_{k}}\in {O}^{c},k\in N$ . If $\stackrel{¯}{\underset{k\to \infty }{lim}}\frac{{7}^{k}}{k!}\underset{\left({l}_{1}\cdots {l}_{k}\right)}{sup}|{\lambda }_{{l}_{1}\cdots {l}_{k}}|=\gamma <\infty$ ,

then the series (5.2) converges to a left ${O}^{c}$ -analytic function $f\left(x\right)$ in the following region

${\Lambda }_{\gamma }=\left\{x\in {R}^{8}:\sqrt{{x}_{0}^{2}+{x}_{i}^{2}}<\frac{1}{\gamma },i=1,2,\cdots ,7\right\}.$

More over, ${\lambda }_{{l}_{1}\cdots {l}_{k}}={\partial }_{{x}_{{l}_{1}}}\cdots {\partial }_{{x}_{{l}_{k}}}f\left(0\right)$ Particularly, if $\underset{\begin{array}{l}\left({l}_{1}\cdots {l}_{k}\right)\\ \text{\hspace{0.17em}}\text{ }k\in N\end{array}}{sup}|{\lambda }_{{l}_{1}\cdots {l}_{k}}|\le C<\infty$ , then $f$ will be a left ${O}^{c}$ -entire function.

Proof. Let

${S}_{N}\left(x\right)=\underset{k=0}{\overset{N}{\sum }}\underset{\left({l}_{1},\cdots ,{l}_{k}\right)}{\sum }{V}_{{l}_{1}\cdots {l}_{k}}\left(x\right){\lambda }_{{l}_{1}\cdots {l}_{k}},\text{\hspace{0.17em}}N\in N.$

For any $x={\sum }_{0}^{7}\text{ }\text{ }{x}_{i}{e}_{i}\in {\Lambda }_{\gamma }$ , there exists ${\gamma }^{\prime }>\gamma$ such that $\sqrt{{x}_{0}^{2}+{x}_{i}^{2}}<\frac{1}{{\gamma }^{\prime }},i=1,2,\cdots ,7$ . Thus

$\begin{array}{l}\underset{x\in {\Lambda }_{{\gamma }^{\prime }}}{\mathrm{sup}}|{S}_{N}\left(x\right)-{S}_{M}\left(x\right)|\\ \le \underset{x\in {\Lambda }_{{\gamma }^{\prime }}}{\mathrm{sup}}\underset{k=M}{\overset{N}{\sum }}\underset{\left({l}_{1},\cdots ,{l}_{k}\right)}{\sum }|{V}_{{l}_{1}\cdots {l}_{k}}\left(x\right)||{\lambda }_{{l}_{1}\cdots {l}_{k}}|\\ \le \underset{x\in {\Lambda }_{{\gamma }^{\prime }}}{\mathrm{sup}}\underset{k=M}{\overset{N}{\sum }}\frac{1}{k!}\underset{{l}_{1},\cdots ,{l}_{k}=1}{\overset{7}{\sum }}|{z}_{{l}_{1}}|\cdots |{z}_{{l}_{k}}||{\lambda }_{{l}_{1}\cdots {l}_{k}}|\\ \le \underset{k=M}{\overset{N}{\sum }}\frac{{7}^{k}}{k!}\frac{1}{{{\gamma }^{\prime }}^{k}}|{\lambda }_{{l}_{1}\cdots {l}_{k}}|\to 0\text{ }\left(\mathrm{inf}\left(M,N\right)\to \infty \right).\end{array}$

From Weierstrass Theorem on octonions [13] and the analyticity of ${V}_{{l}_{1}\cdots {l}_{k}}\left(x\right){\lambda }_{{l}_{1}\cdots {l}_{k}}$ , then there exists a left ${O}^{c}$ -analytic function $f$ in ${\Lambda }_{\gamma }$ such that

$f\left(x\right)=\underset{N\to \infty }{\mathrm{lim}}{S}_{N}\left(x\right)=\underset{k=0}{\overset{\infty }{\sum }}\underset{\left({l}_{1},\cdots ,{l}_{k}\right)}{\sum }{V}_{{l}_{1}\cdots {l}_{k}}\left(x\right){\lambda }_{{l}_{1}\cdots {l}_{k}},$

and the series uniformly converges to $f\left(x\right)$ in each compact subset $K\subset {\Lambda }_{\gamma }$ . Again from the expansion of $f\left(x\right)$ we easily get that ${\lambda }_{{l}_{1}\cdots {l}_{k}}={\partial }_{{x}_{{l}_{1}}}\cdots {\partial }_{{x}_{{l}_{k}}}f\left(0\right)$ .

If $\underset{\begin{array}{l}\left({l}_{1}\cdots {l}_{k}\right)\\ \text{\hspace{0.17em}}\text{ }k\in N\end{array}}{sup}|{\lambda }_{{l}_{1}\cdots {l}_{k}}|\le C<\infty$ , then ${\Lambda }_{\gamma }={R}^{8}$ , since $\underset{k\to \infty }{\mathrm{lim}}\frac{{7}^{k}}{k!}=0$ . Therefore $f$ is a

left ${O}^{c}$ -entire function. ,

Example. Taking ${\lambda }_{{l}_{1}\cdots {l}_{k}}\equiv 1$ for all $k\in N$ in (5.2), then

$\underset{k=0}{\overset{\infty }{\sum }}\underset{\left({l}_{1},\cdots ,{l}_{k}\right)}{\sum }{V}_{{l}_{1}\cdots {l}_{k}}\left(x\right)$ (5.3)

is an $O$ -entire function. In fact, (5.3) is the Taylor expansion of the exponential function $exp\left(x\right)$ as in (3.3). From (3.3) we can find $exp\left(x\right)$ satisfies

$\mathrm{exp}\left(0\right)=1,\text{ }\mathrm{exp}\left(x+y\right)=\mathrm{exp}\left(x\right)\cdot \mathrm{exp}\left(y\right)=\mathrm{exp}\left(y\right)\cdot \mathrm{exp}\left(x\right).$

Corollary 5.3. For any left ${O}^{c}$ -analytic function $f$ , if the coefficients in its Taylor series about the origin satisfy

$\left\{\begin{array}{l}{\partial }_{{x}_{i}^{k}}f\left(0\right)\in C+{e}_{i}C,\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}k\in N,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,7\\ {\partial }_{{x}_{{l}_{1}}}\cdots {\partial }_{{x}_{{l}_{k}}}f\left(0\right)\in C,\text{ }\text{\hspace{0.17em}}\text{otherwise}.\end{array}$ (5.4)

Then $\stackrel{¯}{f}$ is a complex Stein-Weiss conjugate harmonic system.

Proof. From (5.4), we easily obtain that all the conjugates of ${V}_{{l}_{1}\cdots {l}_{k}}\left(x\right){\partial }_{{x}_{{l}_{1}}}\cdots {\partial }_{{x}_{{l}_{k}}}f\left(0\right)$ are complex Stein-Weiss conjugate harmonic systems. Hence by Weierstrass Theorem, $\stackrel{¯}{f}$ 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 [6] we can define the Cauchy-Kowalewski product for any two left ${O}^{c}$ analytic functions f and g in $\Omega$ which containing origin. We let their Taylor expansions be

$f\left(x\right)=\underset{k=0}{\overset{\infty }{\sum }}\underset{\left({l}_{1},\cdots ,{l}_{k}\right)}{\sum }{V}_{{l}_{1}\cdots {l}_{k}}\left(x\right){\partial }_{{x}_{{l}_{1}}}\cdots {\partial }_{{x}_{{l}_{k}}}f\left( 0 \right)$

and

$g\left(x\right)=\underset{t=0}{\overset{\infty }{\sum }}\underset{\left({s}_{1},\cdots ,{s}_{t}\right)}{\sum }{V}_{{s}_{1}\cdots {s}_{t}}\left(x\right){\partial }_{{x}_{{s}_{1}}}\cdots {\partial }_{{x}_{{s}_{t}}}g\left(0\right).$

Then the (left) Cauchy-Kowalewski product of f and g is defined by

$\begin{array}{l}f{\odot }_{L}g\left(x\right)\\ =\underset{k,t=0}{\overset{\infty }{\sum }}\underset{\begin{array}{l}\left({l}_{1},\cdots ,{l}_{k}\right)\\ \left({s}_{1},\cdots ,{s}_{t}\right)\end{array}}{\sum }\left(\underset{i=1}{\overset{7}{\prod }}\frac{\left({n}_{i}+{{n}^{\prime }}_{i}\right)!}{{n}_{i}!\text{ }{{n}^{\prime }}_{i}!}\right){V}_{{l}_{1}\cdots {l}_{k}{s}_{1}\cdots {s}_{t}}\left(x\right)\left({\partial }_{{x}_{{l}_{1}}}\cdots {\partial }_{{x}_{{l}_{k}}}f\left(0\right)\cdot {\partial }_{{x}_{{s}_{1}}}\cdots {\partial }_{{x}_{{s}_{t}}}g\left(0\right)\right),\end{array}$

where ${n}_{i}$ and ${{n}^{\prime }}_{i}$ are the appearing times of i in $\left({l}_{1},\cdots ,{l}_{k}\right)$ and $\left({s}_{1},\cdots ,{s}_{t}\right)$ , respectively.

We have the following relation for the product and the left Cauchy-Kowalewski product between two left ${O}^{c}$ -analytic functions.

Theorem 5.4. Let $f\left(x\right),g\left(x\right)$ be two left ${O}^{c}$ -analytic functions in $\Omega$ which containing origin. If $D\left(f\left(x\right)g\left(x\right)\right)=0$ then

$f\left(x\right)g\left(x\right)=f{\odot }_{L}g\left(x\right).$

Proof. It is easy to see that $f\left(\underset{_}{x}\right)g\left(\underset{_}{x}\right)=f{\odot }_{L}g\left(\underset{_}{x}\right)$ , then by Proposition 2.4 and the analyticity of $f\left(x\right)g\left(x\right)$ and $f{\odot }_{L}g\left(x\right)$ we get $f\left(x\right)g\left(x\right)=f{\odot }_{L}g\left(x\right)$ . ,

Remark. In this paper we study the analyticity of the product of two left ${O}^{c}$ -analytic functions. Theorem 3.2 give some sufficient conditions for the product of two left ${O}^{c}$ -analytic functions is also a left ${O}^{c}$ -analytic function. From Theorem 5.4 we can see that $D\left(f\left(x\right)g\left(x\right)\right)=0$ for two left ${O}^{c}$ -analytic functions $f\left(x\right),g\left(x\right)$ if and only if this product is just equal to their left Cauchy-Kowalewski product. Since $H\subseteq O$ , our result is also true for quaternionic cases.

Funding

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).

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Liao, J. and Wang, J. (2017) The Analyticity for the Product of Analytic Functions on Octonions and Its Applications. Advances in Pure Mathematics, 7, 692-705. doi: 10.4236/apm.2017.712043.

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