Definition:Summation/Inequality/Multiple Indices

Definition

Let $\ds \sum_{0 \mathop \le j \mathop \le n} a_j$ denote the summation of $\tuple {a_0, a_1, a_2, \ldots, a_n}$.

Summands with multiple indices can be denoted by propositional functions in several variables, for example:

$\ds \sum_{0 \mathop \le i \mathop \le n} \paren {\sum_{0 \mathop \le j \mathop \le n} a_{i j} } = \sum_{0 \mathop \le i, j \mathop \le n} a_{i j}$

$\ds \sum_{0 \mathop \le i \mathop \le n} \paren {\sum_{0 \mathop \le j \mathop \le i} a_{i j} } = \sum_{0 \mathop \le j \mathop \le i \mathop \le n} a_{i j}$

Also see

• Results about summations can be found here.

Examples

Sum of Subscripts

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $a$ be an $n$-tuply subscripted variable.

Consider the expression:

$\ds \sum_{\substack {j_1 \mathop + j_2 \mathop + \mathop \cdots \mathop + j_n \mathop = n \\ j_i \mathop \ge j_2 \mathop \ge \mathop \cdots \mathop \ge j_n \mathop \ge 0} } a_{j_1 j_2 \ldots j_n}$

For $n = 5$, this means:

$a_{11111} + a_{21110} + a_{22100} + a_{31100} + a_{32000} + a_{41000} + a_{50000}$

Historical Note

The notation $\sum$ for a summation was famously introduced by Joseph Fourier in $1820$:

Le signe $\ds \sum_{i \mathop = 1}^{i \mathop = \infty}$ indique que l'on doit donner au nombre entier $i$ toutes les valeurs $1, 2, 3, \ldots$, et prendre la somme des termes.

(The sign $\ds \sum_{i \mathop = 1}^{i \mathop = \infty}$ indicates that one must give to the whole number $i$ all the values $1, 2, 3, \ldots$, and take the sum of the terms.)

-- 1820: Refroidissement séculaire du globe terrestre (Bulletin des Sciences par la Société Philomathique de Paris Vol. 3, 7: pp. 58 – 70)

However, some sources suggest that it was in fact first introduced by Euler.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products