Symbols:Pi

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$\pi$

The real number $\pi$ (pronounced pie) is an irrational number (see proof here) whose value is approximately $3.14159\ 26535\ 89793\ 23846\ 2643 \ldots$ This sequence is A000796 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The $\LaTeX$ code for $\pi$ is \pi.

$\Pi_X \left({s}\right)$

Let $X$ be a discrete random variable whose codomain is a subset of $\N = \left\{{0, 1, 2, \ldots}\right\}$.

The probability generating function (p.g.f.) for (or of) $X$ is denoted $\Pi_X \left({s}\right)$ and defined as:

$\Pi_X \left({s}\right) = E \left({s^X}\right)$

where:

The $\LaTeX$ code for $\Pi_X \left({s}\right)$ is \Pi_X \left({s}\right).

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

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

Then the composite is called the product of $\left({a_1, a_2, \ldots, a_n}\right)$, and is written:

$\displaystyle \prod_{j=1}^n a_j = \left({a_1 \times a_2 \times \cdots \times a_n}\right)$

Alternatively:

$\displaystyle \prod_{1 \le j \le n} a_j = \left({a_1 \times a_2 \times \cdots \times a_n}\right)$

If $\Phi \left({j}\right)$ is a propositional function of $j$, then we can write:

$\displaystyle \prod_{\Phi \left({j}\right)} a_j = \text{ The product of all } a_j \text{ such that } \Phi \left({j}\right) \text{ holds}$.

The $\LaTeX$ code for $\displaystyle \prod_{j=1}^n a_j$ is \displaystyle \prod_{j=1}^n a_j.

The $\LaTeX$ code for $\displaystyle \prod_{1 \le j \le n} a_j$ is \displaystyle \prod_{1 \le j \le n} a_j.

The $\LaTeX$ code for $\displaystyle \prod_{\Phi \left({j}\right)} a_j$ is \displaystyle \prod_{\Phi \left({j}\right)} a_j.