Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums

Abstract:

We show that integrals of the form \[ \int _{0}^{1} x^{m}\operatorname {Li}_{p}(x)\operatorname {Li}_{q}(x)dx \quad (m\geq -2, p,q\geq 1) \] and \[ \int _{0}^{1} \frac {\log ^{r}(x) \operatorname {Li}_{p}(x) \operatorname {Li}_{q}(x)}{x}dx\quad (p,q,r\geq 1) \] satisfy certain recurrence relations which allow us to write them in terms of Euler sums. From this we prove that, in the first case for all $m,p,q$ and in the second case when $p+q+r$ is even, these integrals are reducible to zeta values. In the case of odd $p+q+r$, we combine the known results for Euler sums with the information obtained from the problem in this form to give an estimate on the number of new constants which are needed to express the above integrals for a given weight $p+q+r$. The proofs are constructive, giving a method for the evaluation of these and other similar integrals, and we present a selection of explicit evaluations in the last section.