In this paper, by using the method of partial fraction decomposition and integral representations of series, we establish some expressions of series involving harmonic numbers and binomial coefficients in terms of zeta values and harmonic numbers. Furthermore, we can obtain some closed form representations of sums of products of quadratic (or cubic) harmonic numbers and reciprocal binomial coefficients, and some explicit evaluations are given as applications. The given representations are new.

By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and integrals of logarithmic functions. As applications of these relations, we show that multiple zeta values of the form ζ ( 1 ¯ , 1 m − 1 , 1 ¯ , 1 k − 1 ) , ( k , m ∈ N ) $$\begin{array}{} \zeta ( {\bar 1,{{\left\{ 1 \right\}}_{m - 1}},\bar 1,{{\left\{ 1 \right\}}_{k - 1}}} ),\qquad (k,m\in \mathbb{N}) \end{array} $$ for m = 1 or k = 1, and ζ ( 1 ¯ , 1 m − 1 , p , 1 k − 1 ) , ( k , m ∈ N ) $$\begin{array}{} \zeta ( {\bar 1,{{\left\{ 1 \right\}}_{m - 1}},p,{{\left\{ 1 \right\}}_{k - 1}}}),\qquad (k,m\in\mathbb{N}) \end{array} $$ for p = 1 and 2, satisfy certain recurrence relations which allow us to write them in terms of zeta values, polylogarithms and ln 2. Furthermore, we also obtain reductions for certain multiple polylogarithmic values at 1 2 $\begin{array}{} \frac {1}{2} \end{array} $ .