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.

Infinite case

Now consider the case where the fiber of truth of $R$ is infinite.

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.