Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order

In the setting of The Real Numbers: Standing Notation and Background, each natural number being identified with its image in $\mathbb{R}$ under the canonical map, and $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\subseteq\mathbb{R}$. For $n\in\mathbb{N}$, $n!$ is the factorial of $n$, and $0!=1$; for real $y$ and $j\in\mathbb{N}$, $y^{j}$ is the $j$th power of $y$, and $y^{0}=1$; finite products are those of Finite Product Notation.

1. (Multi-indices)¶ A multi-index is a sequence $\alpha=(\alpha_{k})_{k\in\mathbb{N}}$ in $\mathbb{N}_{0}$ for which there is $n\in\mathbb{N}$ with $\alpha_{k}=0$ for every $k>n$; such an $n$ is a length bound for $\alpha$. $\mathcal{A}$ is the set of multi-indices.

2. (Order, factorial, powers and weighted order)¶ For $\alpha\in\mathcal{A}$ with length bound $n$ and a real sequence $s=(s_{k})_{k\in\mathbb{N}}$, the order of $\alpha$, its factorial, the power $s^{\alpha}$ and the weighted order $s\cdot\alpha$ are

$$|\alpha|=\sum_{k=1}^{n}\alpha_{k}\in\mathbb{N}_{0},\qquad\alpha!=\prod_{k=1}^{n}\alpha_{k}!,\qquad s^{\alpha}=\prod_{k=1}^{n}s_{k}^{\alpha_{k}},\qquad s\cdot\alpha=\sum_{k=1}^{n}\alpha_{k}s_{k};$$

they do not depend on the length bound, since passing to a larger length bound adds only terms $\alpha_{k}=0$ and $\alpha_{k}s_{k}=0$ and factors $0!=1$ and $s_{k}^{0}=1$. For $m\in\mathbb{N}_{0}$, $\mathcal{A}_{m}=\{\alpha\in\mathcal{A}:|\alpha|=m\}$.

3. (Unit multi-indices)¶ For $k\in\mathbb{N}$, $\varepsilon_{k}\in\mathcal{A}$ is the multi-index with $k$-th term $1$ and all other terms $0$ (a length bound is $k$). For $\alpha\in\mathcal{A}$ and $k\in\mathbb{N}$ with $\alpha_{k}\ge1$, $\alpha-\varepsilon_{k}$ is the sequence with $k$-th term $\alpha_{k}-1$ and all other terms those of $\alpha$; it is a multi-index, with every length bound of $\alpha$ as a length bound, since $\alpha_{k}-1\in\mathbb{N}_{0}$ for $\alpha_{k}\in\mathbb{N}$.