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

Abstract

Abstract. In recent years, there has been intensive research on the 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 → 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 ⋆-elements ζ ⋆ (s1, . . . , sr) with si ∈ {2, 3} span the vector space generated by multiple zeta values over Q.

On $$q$$ q -analogues of two-one formulas for multiple harmonic sums and multiple zeta star values

“…The generalizations of Theorems 1.1 and 1.2 to arbitrary collections of arguments as well as their limiting cases (q → 1) related to the corresponding results for ordinary multiple zeta values (1), (2) can be found in [13].…”

“…Note that no extension of Apéry's proof leading to the irrationality of a q-analogue of ζ(3) is known. In this respect, formula (13) may be quite helpful: firstly, in view of the fast convergence of the series (13) and secondly, because of the fact that the irrationality proof of ζ(3) as a double series, ζ(2, 1), is known (see [27,28]). …”

Multiple zeta functions and polylogarithms over global function fields

“…Here the notation indicates that the indexes are ordered with non-strict inequalities. Multiple zeta (star) values have been largely studied, see for example, [2,3,8,13,14,15,18,19,20,24,25,26,29,30,31,32,35,41,42,45]. They generally yield simpler identities than multiple zeta values.…”