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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
$$\Delta x^n=(x+1)^nx^n=\sum_{k=1}^n \binom {n}{k} x^{nk}.$$
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References
Bibliography
 Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", AddisonWesley, 1994, 2nd Edition
 Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960
 Bool, George: "A Treatise on the Calculus of Finite Differences", Dover Publications, Inc., 1960