Relations of multiple zeta values and their algebraic expression

By Michael E. Hoffman, Yasuo Ohno. 2000

118 citations recorded by the source corpus.

Abstract We establish a new class of relations, which we call the cyclic sum identities, among the multiple zeta values ζ(k1,…,kl)=∑n1>⋯>nl⩾11/(n1k1⋯nkkl). These identities have an elementary proof and imply the “sum theorem” for multiple zeta values. They also have a succinct statement in terms of “cyclic derivations” as introduced by Rota, Sagan, and Stein. In addition, we discuss the expression of other relations of multiple zeta values via the shuffle and “harmonic” products on the underlying vector space H of the noncommutative polynomial ring Q 〈x,y〉 , and also using an action of the Hopf algebra of quasi-symmetric functions on Q 〈x,y〉 .

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bb9958e5583abecc423254867ffa737c36e57333

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Michael E. Hoffman
2. Li Guo
3. Kurusch Ebrahimi-Fard
4. K. Ihara
5. Jos Vermaseren
6. David John Broadhurst
7. J. Blumlein
8. Gaku Kawashima

In Discrete Mathematics and Combinatorics

1. Michael E. Hoffman
2. K. Ihara
3. Li Guo
4. U. Kühn
5. Henrik Bachmann
6. J. Blumlein
7. Jos Vermaseren
8. David John Broadhurst

In Geometry and Topology

1. Li Guo
2. Kurusch Ebrahimi-Fard
3. U. Kühn
4. Henrik Bachmann
5. J. Okuda
6. K. Ihara
7. David John Broadhurst
8. Jos Vermaseren

Citations in the corpus

• Mixable shuffles, quasi-shuffles and Hopf algebras (2005, background): “Hoffman’s quasi-shuffle product plays a prominent rôle in the recent studies of harmonic functions, quasi-symmetric functions, multiple zeta values [25, 27, 28, 4] (where in special cases it is also called stuffle product or harmonic product) and q-multiple zeta values [5].”
• The algebra of generating functions for multiple divisor sums and applications to multiple zeta values (2013, methodology): “nd the stuffle relations give two different description of the product of two multiple zeta values, albeit in practice there are different methods to prove distinct relations like the cyclic sum identity [HO] or the Zagier-Ohno relation [OZ]. So far we know only one way to write a product of two multipl”
• QUASI-SYMMETRIC FUNCTIONS AND MOD p MULTIPLE HARMONIC SUMS (2004, background): “In fact, there are two distinct multiplications on the set of multiple zeta values, the harmonic (or “stuffle”) product and the shuffle product, coming from the representation of multiple zeta values as iterated series and as iterated integrals respectively (see §4 of [19] for a discussion).”
• A class of relations among multiple zeta values (2007, background): “Ohno’s relation, the cyclic sum formula and the derivation relation [ 7 , 8, 15, 16].”
• On relations for the multiple q-zeta values (2007, background): “, 1] are given by a polynomial of ζ[2], ζ[3], .”
• The Multiple Zeta Value data mine (2009, methodology): “This is done by including the shuffles involving the basic divergent object and breaking down the multiple divergent sums with the stuffle relations as in: H1H0,0,1 = 2H0,0,1,1+H0,1,0,1 +H1,0,0,1 = −H0,0,0,1+H0,0,1,1+H0,1,0,1+H1H0,0,1; (5.4) In the case we useH1 as the only divergent object, this is”
• Multiple zeta values vs. multiple zeta-star values (2011, methodology): “The formulas for MZVs, MZSVs, qMZVs and qMZSVs are established in [13,21,5] and [20], respectively.”