Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums
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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.References
- 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, DOI 10.1137/0519064
- Ankur Basu and Tom M. Apostol, A new method for investigating Euler sums, Ramanujan J. 4 (2000), no. 4, 397–419. MR 1811905, DOI 10.1023/A:1009868016412
- David Borwein and Jonathan M. Borwein, On an intriguing integral and some series related to $\zeta (4)$, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1191–1198. MR 1231029, DOI 10.1090/S0002-9939-1995-1231029-X
- 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, DOI 10.1017/S0013091500019088
- Wenchang Chu, Hypergeometric series and the Riemann zeta function, Acta Arith. 82 (1997), no. 2, 103–118. MR 1477505, DOI 10.4064/aa-82-2-103-118
- 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, DOI 10.1007/BF02724330
- P. J. de Doelder, On some series containing $\psi (x)-\psi (y)$ and $(\psi (x)-\psi (y))^2$ for certain values of $x$ and $y$, J. Comput. Appl. Math. 37 (1991), no. 1-3, 125–141. MR 1136919, DOI 10.1016/0377-0427(91)90112-W
- Philippe Flajolet and Bruno Salvy, Euler sums and contour integral representations, Experiment. Math. 7 (1998), no. 1, 15–35. MR 1618286
- R. Gastmans and W. Troost, On the evaluation of polylogarithmic integrals, Simon Stevin 55 (1981), no. 4, 205–219. MR 647134
- 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. S. Kölbig, Closed expressions for $\int ^{1}_{0}t^{-1}\textrm {log}^{n-1}\ t\log ^{p}(1-t)\,dt$, Math. Comp. 39 (1982), no. 160, 647–654. MR 669656, DOI 10.1090/S0025-5718-1982-0669656-X
- 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
- H. M. Srivastava and Junesang Choi, Series associated with the zeta and related functions, Kluwer Academic Publishers, Dordrecht, 2001. MR 1849375, DOI 10.1007/978-94-015-9672-5
- 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
Additional Information
- Pedro Freitas
- Affiliation: Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
- Email: pfreitas@math.ist.utl.pt
- 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: https://doi.org/10.1090/S0025-5718-05-01747-3
- MathSciNet review: 2137010