linguistic:paraphrasing, stylistic:tone, structural:scholar_edition
Example

Expressing a polynomial in the basis of falling factorials is useful for calculating sums of the polynomial evaluated at consecutive integers.Indeed, the sum of falling factorials with fixed k can expressed as another falling factorial (for k ≠ − 1 {\displaystyle k\neq -1})

This can be proved by induction.

For example, the sum of fourth powers of integers up to n (this time with n included), is:

Here the Stirling numbers can be computed from their definition as the number of partitions of 4 elements into k non-empty unlabeled subsets.

In contrast, the sum ∑ i = 0 n i k {\textstyle \sum _{i=0}^{n}i^{k}} in the standard basis is given by Faulhaber's formula, which in general is more complicated.