Euler-Type Sums Involving Harmonic Numbers and Binomial Coefficients

In this paper, by using iterated integral expression of multiple polylogarithm function, we establish some identities relating multiple harmonic (star) sums and (alternating) multiple zeta values. We then apply these identities and the shuffle relations to evaluate some Euler-type sums involving harmonic numbers and binomial coefficients, such as \(\begin{aligned}&\sum _{n=1}^{\infty }{\frac{\prod _{j=1}^p{H_{n}^{\left( i_j \right) }}}{(2n+1)(n+r)}},\quad \sum _{n=1}^{\infty }{\frac{\prod _{j=1}^p{H_{n}^{\left( i_j \right) }}}{\left( 2n+1 \right) \left( \begin{array}{c} n+k\\ k\\ \end{array} \right) }} \end{aligned}\) $$\begin{array}{cc}& \sum _{n=1}^{\infty }\frac{{\prod }_{j=1}^p{H}_n^{\left(i_j\right)}}{(2n+1)(n+r)},\sum _{n=1}^{\infty }\frac{{\prod }_{j=1}^p{H}_n^{\left(i_j\right)}}{\left(2,n,+,1\right)\left(\begin{array}{c}n+k\\ k\end{array}\right)}\hfill \end{array}$$ and some other forms.