Definition:Power Sum

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Definition

Let $S$ be a finite set of numbers.

Let $p \in \R$ be a real number.


Then the $p$th power sum of $S$ is defined as:

$S_p = \displaystyle \sum_{x \mathop \in S} x^p$


That is, it is the summation of the $p$th powers of all the elements of $S$.


Also defined as

Most sources define $S$ as a set indexed by $\left\{ {1, 2, \ldots n}\right\}$ or $\left\{ {0, 1, \ldots n}\right\}$, and denote $S_p$ as:

$S_p = \displaystyle \sum_{j \mathop = 1}^n {x_j}^p$

or:

$S_p = \displaystyle \sum_{j \mathop = 0}^n {x_j}^p$

accordingly.


Some sources, depending upon the direction they are taking their arguments, restrict $p$ to the set $\Z_{>0}$ of (strictly) positive integers.


Sources