Consider a following integral: \begin{equation} {\mathcal I}^{(q,p)} := \int\limits_0^1 \frac{Li_2(-\xi)}{\xi}\cdot [\log(1-\xi)]^q \cdot [\log(1+\xi)]^p d\xi \end{equation} where $q$ and $p$ are non-negative integers. By expanding parts of the integrand in a series and then integrating term by term we have arrived at the following result: \begin{eqnarray} {\mathcal I}^{(3,0)} = \frac{\pi^6}{180} + \log(2) \cdot \left( \pi^2 \zeta(3) - 12 \zeta(5)\right) + r \end{eqnarray} where the remainder $r$ reads \begin{eqnarray} r&=& 3! \zeta\left( \begin{array}{rrr} 4, & 1, & 1\\ &\frac{1}{2}& \end{array}\right) \\ &=& \frac{3}{2} \sum\limits_{k=1}^\infty \left( \frac{[H_{k-1}]^2 - H_{k-1}^{(2)}}{k^3}\right)\cdot \left( H_{\frac{k}{2}} - H_{\frac{k-1}{2}}\right) \\ &=& -6 \int_0^{-1} \frac{S_{2,3}(-\xi)}{1-\xi} d\xi \end{eqnarray} Here $S_{2,3}()$ is the Nielsen generalized poly-logarithm. Now the question is the following, firstly is it possible to reduce the multivariate zeta function in the remainder to single zeta functions and secondly what is the result for generic values of $q$ and $p$?


Your Answer

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

Browse other questions tagged or ask your own question.