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

Authors

Kh. Hessami Pilehrood (The Fields Institute for Research in Mathematical Sciences, Toronto, Ontario, Canada)

T. Hessami Pilehrood (The Fields Institute for Research in Mathematical Sciences, Toronto, Ontario, Canada)

Jianqiang Zhao (Department of Mathematics, The Bishop’s School, La Jolla, California, U.S.A.)

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

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