TITLE: Any reference for the series expansion of $\Bigr[-\log(1-t)\Bigr]^x$?
QUESTION [6 upvotes]: Any reference that we can find the following $$\Bigr[-\log(1-t)\Bigr]^x = t^x + x t^x \sum_{k=0}^\infty \psi_k(x+k)\,t^{k+1}; \quad \mbox{for all} \, x\in \mathbb R, \, |t|<1$$
where $\psi_k(.)$ are the Stirling Polynomials (Stirling Polynomials).
Thank in advance

REPLY [4 votes]: It should be remarked that the expansion is an immediate consequence of the Lagrange Inversion Formula, and of the definition of the Stirling polynomials via their generating function (as given e.g. in the Wolfram MathWord link you gave). 
Let $f=\sum_{k=1}^\infty f_kt^k $ and $g=\sum_{k=0}^\infty g_kt^k $ be formal power series inverse of each other, with $f_1=g_1=1$; let $k\in\mathbb{N}$, $x\in\mathbb{C}$ and $y\in\mathbb{C}$  with $k+x+y=0$. Denoting as usual $[t^k]F$ the coefficient of $t^k$ in $F$,  the Lagrange Inversion Formula may be written in the form:
$$\frac{1}{x}[t^k]\left( \frac{f}{t}\right)^x+\frac{1}{y}[t^k]\left( \frac{g}{t}\right)^{y}=0$$
or in the equivalent, less symmetric form 
$$[t^k]\left( \frac{f}{t}\right)^x=\frac{x}{x+k}[t^k]\left( \frac{t}{g}\right)^{k+x}.$$
If we take $f(t):= -\log(1-t) $ then its inverse is $g(t)=1-e^{-t}$, so 
$$[t^k]\left( \frac{-\log(1-t)}{t}\right)^x=\frac{x}{x+k}[t^k]\left( \frac{t}{1-e^{-t}}\right)^{k+x}.$$
By the definition of the Stirling polynomials (with $x+k-1$ in place of $x$) 
$$[t^k]\left( {t \over {1-e^{-t}}} \right) ^{x+k}=  { S_k(x+k-1)  \over k!} $$
so we get $$\big(  -\log(1-t) \big)^x=xt^x \sum_{k=0}^\infty \frac{S_k(x+k-1)}{x+k}{t^k \over k!}  $$
which is the expansion you want.
Incidentally,  the expansion
$$\log(-\log(1-t))=\log t-\sum_{k=1}^\infty {S_k(k-1)\over kk!}t^k,$$
can be obtained differentiating w.rto $x$ at $x=0$.