2
$\begingroup$

I have a recollection of seeing an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$. But I don't remember details, and searching my books and the Internet for an hour did not give any results. Could you please remind me this identity (if it exists)?

$\endgroup$
  • 1
    $\begingroup$ Are you sure about $1/4$ and not $1/3$? I am looking at (5) and (7) here; recall that Clausen function is essentially imaginary part of the dilogarithm... $\endgroup$ – Start wearing purple Aug 10 '14 at 23:02
  • $\begingroup$ Definitely $1/4$. IIRC, there was a difference of polygammas of variable order $\nu$ at points $3/4$ and $1/4$ expressed via $\sum$ of polylogarithms depending on $\nu$. $\endgroup$ – Vladimir Reshetnikov Aug 10 '14 at 23:26
  • 1
    $\begingroup$ Closest I can find is the reflection relation for the polygamma function: $$\psi_n(1-z)-(-1)^n \psi_n(z)=(-1)^n \pi \dfrac{d^n}{dz^n}\cot \pi x.$$ Maybe a special case of that can be written in terms of polylogs? $\endgroup$ – Semiclassical Aug 11 '14 at 2:10
5
$\begingroup$

I have not found a source, but I think I reconstructed it correctly: $$\psi^{(n)}\!\left(\tfrac34\right)-\psi^{(n)}\!\left(\tfrac14\right)=(-1)^n\,4^{n+1}\,n!\,\,\Im\operatorname{Li}_{n+1}(i),\ n\in\mathbb N.$$ In the source where I saw it, the imaginary part was probably written as a difference of two polylog terms.

It can be proved using:

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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