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



Sources