Four parametric linear Euler sums

In this paper, firstly, we aim to investigate analytic continuations of altogether four types of parametric linear Euler sums, by using the Euler-Maclaurin summation formula and the Euler-Boole summation formula. Secondly, we show how nicely three shuffle relations among the four parametric linear Euler sums can be derived by using an L-summing formula due to Hassani and Rahimpour. Thirdly, using one of the shuffle relations, we present certain series representations of product of Riemann zeta functions. Some special cases and relevant connections of the results presented here with those involving known identities are also demonstrated and indicated.