Explicit evaluations of extended Euler sums

In this paper, we consider two types of extended Euler sums:Ep,q(k)=∑n=1∞1nq∑r=1kn1rp,Tp,q(k)=∑n=1∞1nq∑r=1⌊n/k⌋1rp,where ⌊x⌋⌊x⌋ is the largest integer ⩽x⩽x. In particular, Ep,q(1)=Tp,q(1) are classical Euler sums. We develop a systematic method to evaluate these extended Euler sums as well as their corresponding alternating sums when the weight p+qp+q is odd.