The relationship between different Euler sums is explored by selecting special Kernel functions. We get a linear recurrence relation between different Euler sums by using CauchyLindelof lemma and Cquchy residue theorem， which is linked to Riemann zeta function. It is further proved that the recurrence relation between different alternating Euler sums is closely related to alternating zeta function under some special conditions， and this relation is still a linear relation. Finally， two general sums are obtained by generalizing the situation of Euler sums.