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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums
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by Pedro Freitas PDF
Math. Comp. 74 (2005), 1425-1440 Request permission


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:
  • 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:
  • MathSciNet review: 2137010