Explicit formulas of alternating multiple zeta star values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $

In a recent paper [ 4 ] , Xu studied some alternating multiple zeta values. In particular, he gave two recurrence formulas of alternating multiple zeta values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $. In this paper, we will give the closed forms representations of $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $ in terms of single zeta values and polylogarithms.