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 | 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.

**[AK]**V. S. Adamchik and K. S. Kölbig,*A definite integral of a product of two polylogarithms*, SIAM J. Math. Anal.**19**(1988), no. 4, 926–938. MR**946652**, 10.1137/0519064**[BA]**Ankur Basu and Tom M. Apostol,*A new method for investigating Euler sums*, Ramanujan J.**4**(2000), no. 4, 397–419. MR**1811905**, 10.1023/A:1009868016412**[BB]**David Borwein and Jonathan M. Borwein,*On an intriguing integral and some series related to 𝜁(4)*, Proc. Amer. Math. Soc.**123**(1995), no. 4, 1191–1198. MR**1231029**, 10.1090/S0002-9939-1995-1231029-X**[BBG]**David Borwein, Jonathan M. Borwein, and Roland Girgensohn,*Explicit evaluation of Euler sums*, Proc. Edinburgh Math. Soc. (2)**38**(1995), no. 2, 277–294. MR**1335874**, 10.1017/S0013091500019088**[C]**Wenchang Chu,*Hypergeometric series and the Riemann zeta function*, Acta Arith.**82**(1997), no. 2, 103–118. MR**1477505****[DD]**A. Devoto and D. W. Duke,*Table of integrals and formulae for Feynman diagram calculations*, Riv. Nuovo Cimento (3)**7**(1984), no. 6, 1–39. MR**781905**, 10.1007/BF02724330**[D]**P. J. de Doelder,*On some series containing 𝜓(𝑥)-𝜓(𝑦) and (𝜓(𝑥)-𝜓(𝑦))² for certain values of 𝑥 and 𝑦*, J. Comput. Appl. Math.**37**(1991), no. 1-3, 125–141. MR**1136919**, 10.1016/0377-0427(91)90112-W**[FS]**Philippe Flajolet and Bruno Salvy,*Euler sums and contour integral representations*, Experiment. Math.**7**(1998), no. 1, 15–35. MR**1618286****[GT]**R. Gastmans and W. Troost,*On the evaluation of polylogarithmic integrals*, Simon Stevin**55**(1981), no. 4, 205–219. MR**647134****[GR]**I. S. Gradshteyn and I. M. Ryzhik,*Table of integrals, series, and products*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR**582453****[K]**K. S. Kölbig,*Closed expressions for ∫¹₀𝑡⁻¹𝑙𝑜𝑔ⁿ⁻¹𝑡log^{𝑝}(1-𝑡)𝑑𝑡*, Math. Comp.**39**(1982), no. 160, 647–654. MR**669656**, 10.1090/S0025-5718-1982-0669656-X**[L]**Leonard Lewin,*Polylogarithms and associated functions*, North-Holland Publishing Co., New York-Amsterdam, 1981. With a foreword by A. J. Van der Poorten. MR**618278****[SC]**H. M. Srivastava and Junesang Choi,*Series associated with the zeta and related functions*, Kluwer Academic Publishers, Dordrecht, 2001. MR**1849375****[Z]**Don Zagier,*Values of zeta functions and their applications*, First European Congress of Mathematics, Vol. II (Paris, 1992) Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 497–512. MR**1341859**

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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:
http://dx.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