Translation of Index Variable of Product
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Theorem
- $\ds \prod_{\map R j} a_j = \prod_{\map R {c \mathop + j} } a_{c \mathop + j} = \prod_{\map R {c \mathop - j} } a_{c \mathop - j}$
where:
- $\ds \prod_{\map R j} a_j$ denotes the product over $a_j$ for all $j$ that satisfy the propositional function $\map R j$
- $c$ is an integer constant which is not dependent upon $j$.
Proof
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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