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Mathematics of Computation

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

Author: Pedro Freitas
Journal: Math. Comp. 74 (2005), 1425-1440
MSC (2000): Primary 33E20; Secondary 11M41
Published electronically: February 14, 2005
MathSciNet review: 2137010
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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

Keywords: Polylogarithms, Euler sums, zeta function
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
Article copyright: © Copyright 2005 American Mathematical Society