Power Series Expansion for nth Power of Logarithm of Reciprocal of 1-z
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Theorem
\(\ds \left({\ln \dfrac 1 {1 - z} }\right)^n\) | \(=\) | \(\ds z^n + \dfrac 1 {n + 1} \left[{ {n + 1} \atop n}\right] z^{n + 1} + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n! \sum_{k \mathop \in \Z} \left[{k \atop n}\right] \frac {z^k} {k!}\) |
where $\ds \left[{k \atop n}\right]$ denotes an unsigned Stirling number of the first kind.
Proof
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Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions: $(26)$