TheoremBase

Let $n\in\mathbb{N}$ . A multi-index of length $n$ is an element

\alpha=(\alpha_1,\dots,\alpha_n)\in(\mathbb{N}\cup\{0\})^n.

That is, a multi-index of length $n$ is an ordered $n$ -tuple of nonnegative integers. The zero multi-index of length $n$ is

0=(0,\dots,0).

For each index $i\in\{1,\dots,n\}$ , the $i$ th standard basis multi-index is

e_i=(0,\dots,0,1,0,\dots,0),

with $1$ in the $i$ th position and $0$ elsewhere. If

\alpha=(\alpha_1,\dots,\alpha_n),\qquad \beta=(\beta_1,\dots,\beta_n)

are multi-indices of length $n$ , we write

\beta\le \alpha

if and only if $\beta_i\le \alpha_i$ for every $i\in\{1,\dots,n\}$ . In that case the difference

\alpha-\beta

is the multi-index of length $n$ defined componentwise by

\alpha-\beta=(\alpha_1-\beta_1,\dots,\alpha_n-\beta_n).

In particular, if $\alpha_i>0$ , then $e_i\le \alpha$ and

\alpha-e_i=(\alpha_1,\dots,\alpha_{i-1},\alpha_i-1,\alpha_{i+1},\dots,\alpha_n).

Multi-Index of Length $n$