# Definition:Product of Differences

## Definition

Let $n \in \Z_{> 0}$ be a strictly positive integer.

Let $\tuple {x_1, x_2, \ldots, x_n}$ be an ordered $n$-tuple of real numbers.

The product of differences of $\tuple {x_1, x_2, \ldots, x_n}$ is defined and denoted as:

$\map {\Delta_n} {x_1, x_2, \ldots, x_n} = \ds \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j}$

When the underlying ordered $n$-tuple is understood, the notation is often abbreviated to $\Delta_n$.

Thus $\Delta_n$ is the product of the difference of all ordered pairs of $\tuple {x_1, x_2, \ldots, x_n}$ where the index of the first is less than the index of the second.