# 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.

## Contents

## Rules

A point with coordinates $(x_1,\dots,x_n)$ in the $n$-dimensional space (real, complex or over any other field $\Bbbk$) 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:
$$
\begin{aligned}
|\a|&=\a_1+\cdots+\a_n\in\Z_+^n,
\\
\a!&=\a_1!\cdots\a_n!\qquad\text{(as usual, }0!=1!=1),
\\
x^\a&=x_1^{\a_1}\cdots x_n^{\a_n}\in \Bbbk[x]=\Bbbk[x_1,\dots,x_n],
\\
\a\pm\b&=(\a_1\pm\b_1,\dots,\a_n\pm\b_n)\in\Z^n,
\end{aligned}
$$
The convention extends for the binomial coefficients ($\a\geqslant\b$ means, quite naturally, that $\a_1\geqslant\b_1,\dots,\a_n\geqslant\b_n$):
$$
\binom{\a}{\b}=\binom{\a_1}{\b_1}\cdots\binom{\a_n}{\b_n}=\frac{\a!}{\b!(\a-\b)!},\qquad \text{if}\quad \a\geqslant\b.
$$
The partial derivative operators are also abbreviated:
$$
\partial_x=\biggl(\frac{\partial}{\partial x_1},\dots,\frac{\partial}{\partial x_n}\biggr)=\partial\quad\text{if the choice of $x$ is clear from context.}
$$
The notation for partial derivatives is also quite natural: for a differentiable function $f(x_1,\dots,x_n)$ of $n$ variables,
$$
\partial^a f=\frac{\partial^{|\a|} f}{\partial x^\a}=\frac{\partial^{\a_1}}{\partial x_1^{\a_1}}\cdots\frac{\partial^{\a_n}}{\partial x_n^{\a_n}}f=\frac{\partial^{|\a|}f}{\partial x_1^{\a_1}\cdots\partial x_n^{\a_n}}.
$$
If $f$ is itself a vector-valued function of dimension $m$, the above partial derivatives are $m$-vectors. The notation
$$
\partial f=\bigg(\frac{\partial f}{\partial x}\bigg)
$$
is used to denote the Jacobian matrix of a function $f$ (in general, only rectangular).

- Caveat

The notation $\a>0$ is ambiguous, especially in mathematical economics, as it may either mean that $\a_1>0,\dots,\a_n>0$, or $0\ne\a\geqslant0$.

## Examples

### Binomial formula

$$ (x+y)^\a=\sum_{0\leqslant\b\leqslant\a}\binom\a\b x^{\a-\b} y^\b. $$

### Leibniz formula for higher derivatives of multivariate functions

$$ \partial^\a(fg)=\sum_{0\leqslant\b\leqslant\a}\binom\a\b \partial^{\a-\b}f\cdot \partial^\b g. $$ In particular, $$ \partial^\a x^\beta=\begin{cases} \frac{\b!}{(\b-\a)!}x^{\b-\a},\qquad&\text{if }\a\leqslant\b, \\ \quad 0,&\text{otherwise}. \end{cases} $$

### Taylor series of a smooth function

If $f$ is infinitely smooth near the origin $x=0$, then its Taylor series (at the origin) has the form $$ \sum_{\a\in\Z_+^n}\frac1{\a!}\partial^\a f(0)\cdot x^\a. $$

### Symbol of a differential operator

If
$$
D=\sum_{|\a|\le d}a_\a(x)\partial^\a
$$
is a linear ordinary differential operator with variable coefficients $a_\a(x)$, then its *principal symbol* is the function of $2n$ variables $S(x,p)=\sum_{|\a|=d}a_\a(x)p^\a$.

**How to Cite This Entry:**

Multi-index notation.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Multi-index_notation&oldid=25758