Definition:Summation/Inequality/Multiple Indices
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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