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](http://mathworld.wolfram.com/StirlingPolynomial.html)).


Thank in advance



Any reference for the series expansion of $\Bigr[-\log(1-t)\Bigr]^x$?
I edit my post to answer Carlo Beenakker's remark and also because I would like to add a reference, possibly more accurate than the two below. Theorem 7.1 p.13 of


A. Adelberg, A finite difference approach to degenerate Bernoulli and Stirling polynomials, Discrete Math. 140 (1995), 1-21,


states that, for $s$ a complex number,
$$\left(\frac{(1+t)^{y}-1}{ty}\right)^{s}=\sum\_{n=0}^{\infty}B\_{n,s}(y)t^{n},$$
where $B\_{n,s}(y)$ is a specific polynomial defined from divided differences of binomial coefficients. Letting $y$ tend to 0 gives
$$(\log(1+t))^{s}=t^{s}\sum\_{n=0}^{\infty}B\_{n,s}(0)t^{n},$$
where it can be check that, with $\psi\_{n}$ the Stirling polynomial,
$$B\_{n,s}(0)=\frac{(-1)^{n}}{n!}\frac{s}{s+n}\psi\_{n}(s+n-1).$$
Hence, the seeked expansion at $t=0$ for a real power of $\log(1+t)$ is given by
$$(\log(1+t))^{s}=t^{s}\sum\_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\frac{s}{s+n}\psi\_{n}(s+n-1)t^{n}.$$




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The formula, written in a different form and with a complex exponent, appears on p.642 of this [paper](http://jmre.dlut.edu.cn/en/ch/reader/create_pdf.aspx?file_no=20120601&flag=1&journal_id=en&year_id=2012) where generalized Stirling functions are used. These generalized Stirling functions are also studied in 


Generalized Stirling Functions of Second Kind and Representations of Fractional Order Differences via Derivatives, P.L. BUTZER, A.A. KILBAS, J. TRUJILLO, Journal of Difference Equations and Applications, 2003 Vol. 9 (5), pp. 503–533


see in particular formula (3.9) there, to be compared with the definition of the Stirling polynomials. The link between the Stirling polynomials in your question and the generalized Stirling functions in these references shouldn't be too complicate to derive.


