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

DOI:
https://doi.org/10.1090/S0025-5718-05-01747-3

Published electronically:
February 14, 2005

MathSciNet review:
2137010

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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