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

We show that integrals of the form \[ ∫ 0 1 x m Li p ⁡ ( x ) Li q ⁡ ( x ) d x ( m ≥ − 2 , p , q ≥ 1 ) ∫ ₀¹ xᵐLiₚ(x)Liq(x)dx (m≥ -2, p,q≥ 1) \] and \[ ∫ 0 1 log r ⁡ ( x ) Li p ⁡ ( x ) Li q ⁡ ( x ) x d x ( p , q , r ≥ 1 ) ∫ ₀¹ {log ʳ(x) Liₚ(x) Liq(x)}{x}dx (p,q,r≥ 1) \] 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