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

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Theorem

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

Then:

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

where:

$\ds {r \brace r - k}$ 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


This theorem requires a proof.
It is unclear exactly how the Stirling numbers are extended to the complex plane. Knuth's exposition is uncharacteristically non-explicit.
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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

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