I have come across the following deceptively simple expression:

$$ H_n^s=\sum_{j=1}^n(-1)^{j-1}\left(\begin{array}{c}n\\j\end{array}\right)j^{-s} $$

We have (using eg mathematica, though probably not difficult to prove): $H_n^0=1$, $H_n^1=H_n$ (the harmonic numbers), expressions involving hypergeometric series with unit argument for integer $s<0$ and involving polygamma functions for integer $s>1$. For fixed $n$ the sum is of course finite. My (closely related) questions are:

Does this reduce to values of a known special function for arbitrary real (or complex) $s$?

What is its asymptotic expansion for large $n$?

Is there an efficient numerical method (avoiding cancellations) of evaluating it for large $n$?

**Edit:** Using Noam's approach I found two more terms that check numerically:

$$ H_n^s=\frac{(\ln n)^s}{\Gamma(s+1)}+\frac{\gamma(\ln n)^{s-1}}{\Gamma(s)}+\frac{6\gamma^2+\pi^2}{12\Gamma(s-1)}(\ln n)^{s-2}+\ldots $$

where $\gamma=0.577\ldots$ is the Euler constant. Further asymptotics very welcome.