◀ ▲ ▶Branches / Combinatorics / Proposition: Difference Operator of Falling Factorial Powers

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 $x\not\in\{1,2,\ldots,|n|,|n|+1\}.$

Table of Contents

Proofs: 1

Mentioned in:

Examples: 1
Proofs: 2 3

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

Github:

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