Can anyone explain me how to write the expression $\displaystyle \prod_{j=1}^n (1+x_j^2)$ where $x_j\in\mathbb R$ with multi-index notation.
Thanks.
Obs: I conjecture I can write this as $$\prod_{j=1}^n (1+x_j^2)=\sum_{\alpha\in \{0, 1\}^n} x^{2\alpha}$$ where $\{0, 1\}^n$ is the cartesian product of $n$-copies of $\{0, 1\}$.
Is this correct?
TERMINOLOGY AND NOTATION:
(i) A multi-index is an element $\alpha\in \mathbb N_0^n$ where $\mathbb N_0:=\mathbb N\cup\{0\}$.
(ii) The length of such an $\alpha$ is defined as $$|\alpha|:=\sum_{j=1}^n \alpha_j.$$
(iii) For $x=(x_1, \ldots, x_n)$ we define $$x^\alpha:=\prod_{j=1}^n x_j^{\alpha_j},$$ and $\displaystyle |x|=\left(\sum_{j=1}^n x_j^2\right)^{1/2}$.
The multi-index notation is very useful for writing expressions involving several variables. For example, we might write the Leibnitz rule for two function $f, g\in C^\infty(\mathbb R^n)$ as $$\partial^\alpha(fg)(x)=\sum_{\beta\leq \alpha}\binom{\alpha}{\beta}(\partial^{\alpha-\beta} f)(x)(\partial^\beta g)(x),$$ where we define $\beta\leq \alpha\Leftrightarrow \beta_j\leq \alpha_j$, $$\binom{\alpha}{\beta}:=\left\{\begin{array}{lcl}\displaystyle\prod_{j=1}^n \binom{\alpha_j}{\beta_j}&\textrm{if}&\beta\leq \alpha\\ 0&&\textrm{otherwise}\end{array}\right.,$$ and $$\partial^\alpha:=\prod_{j=1}^n \partial^{\alpha_j}_j, \ \partial_j:=\frac{\partial}{\partial x_j}$$
How about: $$({\bf1}+i{\bf x})^{\alpha}({\bf1}-i{\bf x})^{\alpha}$$ Where the multi index $\alpha$ is simply $(1,1,\ldots1)$. By definition, this is equals to: $$\prod_{j=1}^n({\bf1}+i{\bf x})_j^{1}({\bf1}-i{\bf x})_j^{1}= \prod_{j=1}^n(1+ix_j)(1-ix_j)=\prod_{j=1}^n(1+x_j^2)$$