Some results on Euler sums

Abstract

In the paper, we develop an approach to evaluation of Euler sums that involve harmonic numbers and alternating harmonic numbers. We give explicit formulae for several classes of Euler sums in terms of Riemann zeta values and prove that the quadratic sums ${S_{{l^2},l}}$ and cubic sums ${S_{{l^3},l}}$ reduce to linear sums and polynomials in zeta values. The approach is based on constructive Power series and Cauchy product computations.