Power of Complex Number as Summation of Stirling Numbers of Second Kind
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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.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
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
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.
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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
- 1992: Donald E. Knuth: Two Notes on Notation (Amer. Math. Monthly Vol. 99: pp. 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: $(59)$
