Definition:Product of Differences

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

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

The product of differences of $\left({x_1, x_2, \ldots, x_n}\right)$ is defined and denoted as:
 * $\displaystyle \Delta_n \left({x_1, x_2, \ldots, x_n}\right) = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_i - x_j}\right)$

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 $\left({x_1, x_2, \ldots, x_n}\right)$ where the index of the first is less than the index of the second.

Also see

 * Definition:Vandermonde Polynomial