Looking for an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$ 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)?

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:


*

*http://mathworld.wolfram.com/PolygammaFunction.html, (5)

*http://dlmf.nist.gov/25.11, 25.11.31

*Gradshteyn—Ryzhik, 3.523

*http://mathworld.wolfram.com/LegendresChi-Function.html, (2), (4)

