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

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## 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.

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## Additional Information

**Pedro Freitas**- Affiliation: Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
- Email: pfreitas@math.ist.utl.pt
- Received by editor(s): August 28, 2003
- Received by editor(s) in revised form: March 9, 2004
- Published electronically: February 14, 2005
- Additional Notes: This author was partially supported by FCT, Portugal, through program POCTI
- © Copyright 2005 American Mathematical Society
- Journal: Math. Comp.
**74**(2005), 1425-1440 - MSC (2000): Primary 33E20; Secondary 11M41
- DOI: https://doi.org/10.1090/S0025-5718-05-01747-3
- MathSciNet review: 2137010