∆ᵏ-constant Sequence

Overview

Let $(a_n)$ be a sequence. We define the $k$th differences of $(a_n)$ recursively:

• The $0$th differences of $(a_n)$ is $(a_n)$.
• The $k$th differences of $(a_n)$ is the sequence given by subtracting consecutive terms of the $(k-1)st$ differences of $(a_n)$.

A sequence is said to be ${\Delta }^k$-constant if the $k$th differences are constant.

The closed formula for a sequence will be a degree $k$ polynomial if and only if the sequence is ${\Delta }^k$-constant.

This is the discrete analog to (continuous) derivatives of polynomials.

Bibliography

• Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.