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

and

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 and in the second case when is even, these integrals are reducible to zeta values. In the case of odd , 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 .

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

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

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