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# Communications in Number Theory and Physics

## Volume 10 (2016)

### Number 4

### On $q$-analogs of some families of multiple harmonic sums and multiple zeta star value identities

Pages: 805 – 832

DOI: https://dx.doi.org/10.4310/CNTP.2016.v10.n4.a4

#### Authors

#### 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 results of the paper (Theorems 1.4 and 5.4) are 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 $\zeta^{\star}(s_1 , \dotsc , s_r)$ with $s_i \in \lbrace 2, 3 \rbrace$ span the vector space generated by multiple zeta values over $\mathbb{Q}$.

#### Keywords

multiple harmonic sums, multiple zeta values, multiple zeta star values, Euler sums

#### 2010 Mathematics Subject Classification

11B65, 11M32

Published 3 April 2017