On the convolutions of sums of multiple zeta(-star) values of height one

In this paper, we investigate the sums of multiple zeta(-star) values of height one: Z ± ( n ) = ∑ a + b = n ( ± 1 ) b ζ ( { 1 } a , b + 2 ) , Z ± ⋆ ( n ) = ∑ a + b = n ( ± 1 ) b ζ ⋆ ( { 1 } a , b + 2 ) . In particular, we prove that the weighted sum ∑ 0 ≤ m ≤ p m : even ∑ ∣ α ∣ = p + 3 2 α m + 1 + 1 ζ ( α 0 , α 1 , … , α m , α m + 1 + 1 ) can be evaluated through the convolution of Z - ( m ) and Z + ( n ) with m + n = p .