The falling factorial powers allow together with the difference operator a similarly simple compact expression as the nth powers under differentiation.

Proposition: Difference Operator of Falling Factorial Powers

Let $x\in \mathbb C$ be a complex number. Then the difference operator of the falling factorial power $x^{\underline n}$ equals

$$\Delta x^{\underline n}=nx^{\underline {n-1}}$$ for all integers $n\in\mathbb Z.$ If $n\le 0$, we also require $x\not\in\{1,2,\ldots,|n|,|n|+1\}.$

Proofs: 1

Examples: 1
Proofs: 2 3

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  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