Translation of Index Variable of Summation
Theorem
Let $R: \Z \to \set {\T, \F}$ be a propositional function on the set of integers.
Let $\ds \sum_{\map R j} a_j$ denote a summation over $R$.
Then:
- $\ds \sum_{\map R j} a_j = \sum_{\map R {c \mathop + j} } a_{c \mathop + j} = \sum_{\map R {c \mathop - j} } a_{c \mathop - j}$
where $c$ is an integer constant which is not dependent upon $j$.
Corollary
- $\ds \sum_{j \mathop = m}^n a_j = \sum_{j \mathop = m + c}^{n + c} a_{j - c}$
where $c$ is an integer constant which is not dependent upon $j$.
Proof
First we consider the case where the fiber of truth of $R$ is finite.
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Infinite case
Now consider the case where the fiber of truth of $R$ is infinite.
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Examples
- $\ds \sum_{1 \mathop \le j \mathop \le n} a_j = \sum_{1 \mathop \le j - 1 \mathop \le n} a_{j - 1} = \sum_{2 \mathop \le j \mathop \le n + 1} a_{j - 1}$
Also known as
This operation is often referred to as Adjustment of Indices.
Some expositions, when invoking this result, refer to it as an instance of Permutation of Indices of Summation, otherwise known as permutation of the range.
Also see
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