# Singularity analysis: logarithmic scale with general alpha and integral beta (e.g. harmonic numbers)

I'm trying to do singularity analysis to get coefficient asymptotics on functions of the following type:

$$(1-z)^{-\alpha} \log^k \frac{1}{1-z}$$

where $\alpha \notin \mathbb{Z}_{\leq 0}$ general and

$k \in \mathbb{Z}_{\geq 0}$ integral.

The book "Analytic combinatorics" from Flajolet and Sedgewick says that for this special case there exists an complete asymptotic expansion of the form

$$[z^n] (1-z)^{-\alpha} \log^k \frac{1}{1-z} \sim \frac{n^{\alpha-1}}{\Gamma(\alpha)} \left( E_0(\log n) + \frac{E_1(\log n)}{n} + \dots \right)$$

where $E_j$ are polynomials of degree $k$ (see http://algo.inria.fr/flajolet/Publications/AnaCombi/book.pdf page 387, page 403 in pdf).

How can i calculate this $E_j$ explicitly to get the complete asymptotic expansion?

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