Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
2  
Similar question: math.stackexchange.com/questions/203125/… –  Gerry Myerson Sep 27 '12 at 12:18
add comment

2 Answers 2

up vote 4 down vote accepted

Your function can be represented in terms (and is actually quite close to the definition) of the q-Polygamma function. See equation (2).

share|improve this answer
    
Even better, see the "Lambert series" (7) on the same page. –  GEdgar Sep 27 '12 at 14:44
1  
@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
add comment

Mathematica gives

$$f(z)=-\frac{\text{Log}[2]+2 \text{Log}\left[2^{-z}\right]+2 \text{QPolyGamma}\left[0,-\frac{\text{Log}\left[2^{-z}\right]}{\text{Log}[2]},2\right]}{\text{Log}[4]}$$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.