Falling Factorial of Complex Number as Summation of Unsigned Stirling Numbers of First Kind/Proof

Theorem

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

Then:

$z^{\underline r} = \ds \sum_{k \mathop = 0}^m{r \brack r - k} \paren {-1}^k z^{r - k} + \map \OO {z^{r - m - 1} }$

where:

$\ds {r \brack r - k}$ denotes the extension of the unsigned Stirling numbers of the first kind to the complex plane

$z^{\underline r}$ denotes $z$ to the $r$ falling

$\map \OO {z^{r - m - 1} }$ denotes big-$\OO$ notation.

Proof

This theorem requires a proof. In particular: It is unclear exactly how the Stirling numbers are extended to the complex plane. Knuth's exposition is uncharacteristically non-explicit. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $65$