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

We show that integrals of the form \ ∫ 0 1 x m Li p ⁡ (x) Li q ⁡ (x) d x (m ≥ − 2, p, q ≥ 1) ₀^1 x^m Li₏ (x) Liₐ (x) dx (m -2, p, q 1) \ and \ ∫ 0 1 log r ⁡ (x) Li p ⁡ (x) Li q ⁡ (x) x d x (p, q, r ≥ 1) ₀^1 ^{r (x) Li₏ (x) Liₐ (x) }xdx (p, q, r 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 m, p, q and in the second case when p + q + r <mml: annotation encoding="application/x