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

Abstract

We show that integrals of the form \[ \dint_{0}^{1} x^{m}{\rm Li}_{p}(x){\rm Li}_{q}(x)dx, (m\geq -2, p,q\geq 1) \] and \[ \dint_{0}^{1} \frac{\ds \log^{r}(x){\rm Li}_{p}(x){\rm Li}_{q}(x)}{\ds x}dx, (p,q,r\geq 1) \] 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 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 {\it 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.