Euler sums and integrals of polylogarithm functions

This paper develops an approach to evaluation of Euler sums and integrals of polylogarithm functions. The approach is based on simple Cauchy product formula computations. Using the approach, some relationships between Euler sums and integrals of polylogarithm functions are established. A kind of seven, eight and nine order sums of Euler sums are obtained. Furthermore, we give explicit formula for several classes of Euler sums and integrals of polylogarithm functions in terms of Riemann zeta values.