A Family of Multiple Harmonic Sum and Multiple Zeta Star Value Identities

Abstract

In this paper we present a new family of identities for multiple harmonic sums which generalize a recent result of Hessami Pilehrood et al. [7]. We then apply it to obtain a family of identities relating multiple zeta star values to alternating Euler sums. In such a typical identity the entries of the multiple zeta star values consist of blocks of arbitrarily long 2-strings separated by positive integers greater than two while the largest depth of the alternating Euler sums depends only on the number of 2-string blocks but not on their lengths.1991 Mathematics Subject Classification. 11A07, 11M32, 11B65, 11B68.

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

“…Moreover, in all the index sets appearing on the right hand side above, the third components still satisfy (17).…”

“…Summing the multiple sum in the above over k 0 by (11) and noticing that for (p; p; (17), the first component ≈ p 1 can take only values −1 and 0, we obtain with the help of (7) that…”

“…Recently, the first two authors proved a q-analog of the Two-one formula in [10]. Our original goal of this paper was to provide further analogs of the identities contained in [17,22]. However, we have achieved much more because we can now actually treat arbitrary q-MZSV and express it in terms of q-analog Euler sums (of the non-star version).…”

“…In [17,22] Linebarger and the third author obtained many families of identities involving both MHS and MZSV after getting inspiration from [11]. In particular, the third author proved the following so-called Two-one formula conjectured by Ohno and Zudilin in [18]:…”

“…It generalizes the Two-one formula and many other 2-c-2-c, 2-1-2-c, 2-c-2-1 formulas with c ≥ 3, proved by the third author [17,22]. A q-analog of Theorem 1.4 is given in Section 5, which is Theorem 5.4.…”