Definition:Continued Product/Propositional Function

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Definition

Let $\struct {S, \times}$ be an algebraic structure where the operation $\times$ is an operation derived from, or arising from, the multiplication operation on the natural numbers.

Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$.


Let $\map R j$ be a propositional function of $j$.

Then we can write:

$\ds \prod_{\map R j} a_j = \text { the product of all $a_j$ such that $\map R j$ holds}$.


If more than one propositional function is written under the product sign, they must all hold.


Such an operation on an ordered tuple is known as a continued product.


Note that the definition by inequality form $1 \le j \le n$ is a special case of such a propositional function.

Also note that the definition by index form $\ds \prod_{j \mathop = 1}^n$ is merely another way of writing $\ds \prod_{1 \mathop \le j \mathop \le n}$.

Hence all instances of a continued product can be expressed in terms of a propositional function.


Iverson's Convention

Let $\ds \prod_{\map R j} a_j$ be the continued product over all $a_j$ such that $j$ satisfies $R$.


This can also be expressed:

$\ds \prod_{j \mathop \in \Z} a_j^{\sqbrk {\map R j} }$

where $\sqbrk {\map R j}$ is Iverson's convention.


Multiplicand

The set of elements $\set {a_j \in S}$ is called the multiplicand.


Notation

The sign $\ds \prod$ is called the product sign and is derived from the capital Greek letter $\Pi$, which is $\mathrm P$, the first letter of product.


Also see

  • Results about continued products can be found here.


Historical Note

The originally investigation into the theory of infinite products was carried out by Leonhard Paul Euler.


Sources

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