In recent years, there has been intensive research on the ${ \mathbb Q}$-linear relations between multiple zeta (star) values. In this paper, we prove many families of identities involving the $q$-analog of these values, from which we can always recover the corresponding classical identities by taking $q\to 1$. The main result of the paper is the duality relations between multiple zeta star values and Euler sums and their $q$-analogs, which are generalizations of the Two-one formula and some multiple harmonic sum identities and their $q$-analogs proved by the authors recently. Such duality relations lead to a proof of the conjecture by Ihara et al. that the Hoffman $\star$-elements $\zeta^{ \star}(s_1,\dots,s_r)$ with $s_i\in\{ 2,3\}$ span the vector space generated by multiple zeta values over ${ \mathbb Q}$.
Categories: math.NT