Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums

Details Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums

2004-06-21

arxiv.org/abs/math/0406401v1

Math. Comp. 74 (2005), 1425-1440.

Mathematics

Classical Analysis and ODEs

17 pages

Scientific paper

We show that integrals of the form \[ \dint_{0}^{1} x^{m}{\rm Li}_{p}(x){\rm Li}_{q}(x)dx, (m\geq -2, p,q\geq 1) \] and \[ \dint_{0}^{1} \frac{\ds \log^{r}(x){\rm Li}_{p}(x){\rm Li}_{q}(x)}{\ds x}dx, (p,q,r\geq 1) \] 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 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 {\it 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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