# Can $\sum_{k=0}^{\infty}\frac{1}{2^{z+k}-1}$ be expressed in a more compact form?

I'd like to have a more compact form for

$$f(z)=\sum_{k=0}^{\infty}\frac{1}{2^{z+k}-1}$$

Could anyone devise something?

Thanks.

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Similar question: math.stackexchange.com/questions/203125/… –  Gerry Myerson Sep 27 '12 at 12:18

@GEdgar I'm not sure the Lambert series is enough to write the OP's function for every $z.$ –  Ragib Zaman Sep 27 '12 at 14:53
$$f(z)=-\frac{\text{Log}[2]+2 \text{Log}\left[2^{-z}\right]+2 \text{QPolyGamma}\left[0,-\frac{\text{Log}\left[2^{-z}\right]}{\text{Log}[2]},2\right]}{\text{Log}[4]}$$