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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

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

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References

Bibliography

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