I am trying to find a closed expression for summations of the following form:
$\bar\Gamma_{n}(s)=\sum_{t=0}^{\infty} 2^{-ts}t^n$
I have managed to find closed expressions for $\bar\Gamma_{\{0,1,2\}}$ where obviously $0$ is trivial, 1 isn't to complicated but as soon as i got on to 2 it was unbareable.
$\bar\Gamma_{0}(s)=\frac{2^{-s}}{1-2^{-s}}$
$\bar\Gamma_{1}(s)=\frac{2^{-s}}{(1-2^{-s})^2}$
$\bar\Gamma_{2}(s)=\frac{2^{-s}(1+2^{-s})}{(1-2^{-s})^3}$
can someone help me find a closed form for $\bar\Gamma_{n}?$
$$ \bar{\Gamma}_n=(-\ln 2)^{-n}\frac{d^n}{ds^n}\sum_{t=0}^{\infty} 2^{-ts} =(-\ln 2)^{-n}\frac{d^n}{ds^n}\left(\frac{1}{1-2^{-s}}\right). $$