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29
Multiple Zeta Values At NonPositive Integers
, 1999
"... Values of EulerZagier's multiple zeta function at nonpositive integers are studied, especially at (0; 0; : : : ; n) and ( n; 0; : : : ; 0). Further we prove a symmetric formula among values at nonpositive integers. ..."
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Values of EulerZagier's multiple zeta function at nonpositive integers are studied, especially at (0; 0; : : : ; n) and ( n; 0; : : : ; 0). Further we prove a symmetric formula among values at nonpositive integers.
Resolution of some open problems concerning multiple zeta values of arbitrary depth
 Compositio Mathematica
"... Abstract. We prove some new evaluations for multiple polylogarithms of arbitrary depth. The simplest of our results is a multiple zeta evaluation one order of complexity beyond the wellknown BroadhurstZagier formula. Other results we provide settle three of the remaining outstanding conjectures of ..."
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Abstract. We prove some new evaluations for multiple polylogarithms of arbitrary depth. The simplest of our results is a multiple zeta evaluation one order of complexity beyond the wellknown BroadhurstZagier formula. Other results we provide settle three of the remaining outstanding conjectures of Borwein, Bradley, and Broadhurst [4, 5]. A complete treatment of a certain arbitrary depth class of periodic alternating unit Euler sums is also given. 1 Research partially supported by NSF grant DMS9705782. 2
Algebraic aspects of multiple zeta values
 in ”Zeta Functions, Topology and Quantum Physics
, 2005
"... Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves “coding ” the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values ca ..."
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Cited by 16 (2 self)
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Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves “coding ” the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values can then be thought of as defining a map ζ: H 0 → R from a graded rational vector space H 0 generated by the “admissible words ” of the noncommutative polynomial algebra Q〈x,y〉. Now H 0 admits two (commutative) products making ζ a homomorphism–the shuffle product and the “harmonic ” product. The latter makes H 0 a subalgebra of the algebra QSym of quasisymmetric functions. We also discuss some results about multiple zeta values that can be stated in terms of derivations and cyclic derivations of Q〈x,y〉, and we define an action of QSym on Q〈x,y 〉 that appears useful. Finally, we apply the algebraic approach to relations of finite partial sums of multiple zeta value series. 1
Partition identities for the multiple zeta function, to appear
 in Zeta Functions, Topology, and Physics, Kinki University Mathematics Seminar Series, Developments in Mathematics. http://arXiv.org/abs/math.CO/0402091
"... Abstract. We define a class of expressions for the multiple zeta function, and show how to determine whether an expression in the class vanishes identically. The class of such identities, which we call partition identities, is shown to coincide with the class of identities that can be derived as a c ..."
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Cited by 16 (9 self)
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Abstract. We define a class of expressions for the multiple zeta function, and show how to determine whether an expression in the class vanishes identically. The class of such identities, which we call partition identities, is shown to coincide with the class of identities that can be derived as a consequence of the stuffle multiplication rule for multiple zeta values. 1.
A qanalog of Euler’s decomposition formula for the double zeta function
, 2005
"... The double zeta function was first studied by Euler in response to a letter from Goldbach in 1742. One of Euler’s results for this function is a decomposition formula, which expresses the product of two values of the Riemann zeta function as a finite sum of double zeta values involving binomial coef ..."
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Cited by 11 (2 self)
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The double zeta function was first studied by Euler in response to a letter from Goldbach in 1742. One of Euler’s results for this function is a decomposition formula, which expresses the product of two values of the Riemann zeta function as a finite sum of double zeta values involving binomial coefficients. In this note, we establish a qanalog of Euler’s decomposition formula. More specifically, we show that Euler’s decomposition formula can be extended to what might be referred to as a “double qzeta function ” in such a way that Euler’s formula is recovered in the limit as q tends to 1.
The Hopf algebra structure of multiple harmonic sums
, 2004
"... Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of their properties that can be exploited in calculations. ..."
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Cited by 10 (0 self)
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Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of their properties that can be exploited in calculations.
On MordellTornheim sums and multiple zeta values
, 2010
"... We prove that any MordellTornheim sum with positive integer arguments can be expressed as a rational linear combination of multiple zeta values of the same weight and depth. By a result of Tsumura, it follows that any MordellTornheim sum with weight and depth of opposite parity can be expressed ..."
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Cited by 10 (4 self)
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We prove that any MordellTornheim sum with positive integer arguments can be expressed as a rational linear combination of multiple zeta values of the same weight and depth. By a result of Tsumura, it follows that any MordellTornheim sum with weight and depth of opposite parity can be expressed as a rational linear combination of products of multiple zeta values of lower depth.
SHUFFLE PRODUCTS FOR MULTIPLE ZETA VALUES AND PARTIAL FRACTION DECOMPOSITIONS OF ZETAFUNCTIONS OF ROOT Systems
, 2009
"... The shuffle product plays an important role in the study of multiple zeta values. This is expressed in terms of multiple integrals, and also as a product in a certain noncommutative polynomial algebra over the rationals in two indeterminates. In this paper, we give a new interpretation of the shuf ..."
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Cited by 7 (4 self)
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The shuffle product plays an important role in the study of multiple zeta values. This is expressed in terms of multiple integrals, and also as a product in a certain noncommutative polynomial algebra over the rationals in two indeterminates. In this paper, we give a new interpretation of the shuffle product. In fact, we prove that the procedure of shuffle products essentially coincides with that of partial fraction decompositions of multiple zeta values of root systems. As an application, we give a proof of extended double shuffle relations without using Drinfel’d integral expressions for multiple zeta values. Furthermore, our argument enables us to give some functional relations which include double shuffle relations.
Signed qanalogs of Tornheim’s double series
 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 136, 2689–2698, 2008. MARKUS KUBA, INSTITUT FÜR DISKRETE MATHEMATIK UND GEOMETRIE, TECHNISCHE UNIVERSITÄT WIEN, WIEDNER HAUPTSTR. 810/104, 1040
, 2008
"... We introduce signed qanalogs of Tornheim’s double series and evaluate them in terms of double qEuler sums. As a consequence, we provide explicit evaluations of signed and unsigned Tornheim double series and correct some mistakes in the literature. ..."
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We introduce signed qanalogs of Tornheim’s double series and evaluate them in terms of double qEuler sums. As a consequence, we provide explicit evaluations of signed and unsigned Tornheim double series and correct some mistakes in the literature.