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22
Massive 3loop Feynman Diagrams Reducible to SC
 Primitives of Algebras of the Sixth Root of Unity. http://lanl.arxiv.org/abs/hepth/9803091
, 1998
"... Abstract In each of the 10 cases with propagators of unit or zero mass, the nite part of the scalar 3loop tetrahedral vacuum diagram is reduced to 4letter words in the 7letter alphabet of the 1forms: = dz=z and!p: = dz= (,p, z), where is the sixth root of unity. Three diagrams yield only ( 3!0) ..."
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Cited by 56 (8 self)
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Abstract In each of the 10 cases with propagators of unit or zero mass, the nite part of the scalar 3loop tetrahedral vacuum diagram is reduced to 4letter words in the 7letter alphabet of the 1forms: = dz=z and!p: = dz= (,p, z), where is the sixth root of unity. Three diagrams yield only ( 3!0) = 1
Combinatorial aspects of multiple zeta values
 Electr. J. Comb
, 1998
"... 1 the electronic journal of combinatorics 5 (1998), #R38 2 Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle p ..."
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Cited by 42 (8 self)
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1 the electronic journal of combinatorics 5 (1998), #R38 2 Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle product rule allows the possibility of a combinatorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with certain repeated arguments. We also prove a similar cyclic sum identity. Finally, we present extensive computational evidence supporting an infinite family of conjectured MZV identities that simultaneously generalize the Zagier identity. 1
Experimental Mathematics: Recent Developments and Future Outlook
 CECM PREPRINT 99:143] FFL J.M. BORWEIN AND P.B. BORWEIN, &QUOT;CHALLENGES FOR MATHEMATICAL COMPUTING,&QUOT; COMPUTING IN SCIENCE & ENGINEERING, 2001. [CECM PREPRINT 01:160
, 2000
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Emerging Tools for Experimental Mathematics
 American Mathematical Monthly
, 1999
"... This paper is not a tutorial on how lattice basis reduction algorithms such as LLL or PSLQ actually work; rather, we discuss some of the ways these tools can be used to generate conjectures, and for that, a detailed understanding of the underlying algorithms is not necessary. We do hope, however, to ..."
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Cited by 22 (10 self)
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This paper is not a tutorial on how lattice basis reduction algorithms such as LLL or PSLQ actually work; rather, we discuss some of the ways these tools can be used to generate conjectures, and for that, a detailed understanding of the underlying algorithms is not necessary. We do hope, however, to convey some appreciation of their power. We begin with some warmup examples, using the Inverse Symbolic Calculator (ISC); http:// www.cecm.sfu.ca/ MRG/ INTERFACES.html. The basic idea is simple: given the first few decimal digits of some real number, we want the ISC to guess a formula for what it `really' is. For example, if we input K 1 = 3:14626436994198, and click on simple lookup
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
Special values of generalized logsine integrals
"... We study generalized logsine integrals at special values. At π and multiples thereof explicit evaluations are obtained in terms of multiple polylogarithms at ±1. For general arguments we present algorithmic evaluations involving polylogarithms at related arguments. In particular, we consider logsi ..."
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Cited by 11 (7 self)
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We study generalized logsine integrals at special values. At π and multiples thereof explicit evaluations are obtained in terms of multiple polylogarithms at ±1. For general arguments we present algorithmic evaluations involving polylogarithms at related arguments. In particular, we consider logsine integrals at π/3 which evaluate in terms of polylogarithms at the sixth root of unity. An implementation of our results for the computer algebra systems Mathematica and SAGE is provided. 1.
Determinations of rational Dedekindzeta invariants of hyperbolic manifolds and Feynman knots and links
, 1998
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Logsine Evaluations of Mahler measures
, 2010
"... We provide evaluations of several recently studied higher and multiple Mahler measures using logsine integrals. We then give additional related evaluations that are made possible by our methods. We also explore related generating functions for the logsine integrals. ..."
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Cited by 10 (7 self)
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We provide evaluations of several recently studied higher and multiple Mahler measures using logsine integrals. We then give additional related evaluations that are made possible by our methods. We also explore related generating functions for the logsine integrals.
Computation and theory of extended MordellTornheimWitten sums
, 2012
"... We consider some fundamental generalized Mordell–Tornheim–Witten (MTW) zetafunction values along with their derivatives, and explore connections with multiplezeta values (MZVs). To achieve this, we make use of symbolic integration, high precision numerical integration, and some interesting combin ..."
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Cited by 8 (4 self)
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We consider some fundamental generalized Mordell–Tornheim–Witten (MTW) zetafunction values along with their derivatives, and explore connections with multiplezeta values (MZVs). To achieve this, we make use of symbolic integration, high precision numerical integration, and some interesting combinatorics and specialfunction theory. Our original motivation was to represent unresolved constructs such as Eulerian loggamma integrals. We are able to resolve all such integrals in terms of a MTW basis. We also present, for a substantial subset of MTW values, explicit closedform expressions. In the process, we significantly extend methods for highprecision numerical computation of polylogarithms and their derivatives with respect to order.
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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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.