New relations among Euler sums of even weight

Abstract

IIn this paper, we shall consider different kinds of Euler sums which are related to Ramanujan's constant $G(1)$. We develop new relations among these Euler sums and classical Euler sums of even weight. In particular, from these relations we give explicit evaluations of $$ G_{1, 2n+1} = \sum_{k=0}^{\infty} \frac{1}{(2k+1)^{2n+1}} \sum_{j=1}^{k} \frac{1}{j}. $$