Proposition: Difference Operator of Powers

Let $n\ge 1$ be a positive integer and let $x\in \mathbb C$ be a complex number. Then the difference operator of the nth power $x^n$ equals

$$\Delta x^n=(x+1)^n-x^n=\sum_{k=1}^n \binom {n}{k} x^{n-k}.$$

Proofs: 1

Definitions: 1

Thank you to the contributors under CC BY-SA 4.0!




  1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition
  2. Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960
  3. Bool, George: "A Treatise on the Calculus of Finite Differences", Dover Publications, Inc., 1960