# Power of Complex Number as Summation of Stirling Numbers of Second Kind

Jump to navigation
Jump to search

## Contents

## Theorem

Let $z \in \C$ be a complex number whose real part is positive.

Then:

- $z^r = \displaystyle \sum_{k \mathop \in \Z} {r \brace r - k} z^{\underline {r - k} }$

where:

- $\displaystyle {r \brace r - k}$ denotes the extension of the Stirling numbers of the second kind to the complex plane
- $z^{\underline {r - k} }$ denotes $z$ to the $r - k$ falling.

## Proof

## Historical Note

Donald E. Knuth acknowledges the work of Benjamin Franklin Logan in his *The Art of Computer Programming: Volume 1: Fundamental Algorithms, 3rd ed.* of $1997$.

## Sources

- 1992: Donald E. Knuth:
*Two Notes on Notation*(*Amer. Math. Monthly***Vol. 99**: 403 – 422) www.jstor.org/stable/2325085 - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(59)$