On $q$-Analogs of Some Families of Multiple Harmonic Sum and Multiple Zeta Star Value Identities

Abstract

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 $\mathbb{Q}$.

The abstract and references are updated. Two new theorems, Theorem 1.4, Theorem 5.4, and Corollary 1.6 are added

References in corpus (2)

• Identity Families of Multiple Harmonic Sums and Multiple Zeta (Star) Values
• On $q$-analogues of two-one formulas for multiple harmonic sums and multiple zeta star values

Cited by in corpus (1)

• Multiple zeta values and Euler sums