Multi-index notation

$\def\a{\alpha}$ $\def\b{\beta}$

An abbreviated form of notation in analysis, imitating the vector notation by single letters rather than by listing all vector components.

Monomials

A point with coordinates $(x_1,\dots,x_n)$ in the $n$-dimensional space (real, complex or over any other field) is denoted by $x$. For a multiindex $\a=(\a_1,\dots,\a_n)\in\Z_+^n$ the expression $x^\a$ denotes the product, $x_\a=x_1^{\a_1}\cdots x_n^{\a_n}$. Other expressions related to multiindices are expanded as follows: $$ \begin{aligned} |\a|&=\a_1+\cdots+\a_n\in\Z_+^n, \\ \a!&=\a_1!\cdots\a_n!\qquad\text{(as usual,}0!=1!=1), \\ x^\a&=x_1^{\a_1}\cdots x_n^{\a_n}, \\ \a\pm\b&=(\a_1\pm\b_1,\dots,\a_n\pm\b_n)\in\Z^n, \end{aligned} $$