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51
LOOPERASED WALKS AND TOTAL POSITIVITY
, 2000
"... We consider matrices whose elements enumerate weights of walks in planar directed weighted graphs (not necessarily acyclic). These matrices are totally nonnegative; more precisely, all their minors are formal power series in edge weights with nonnegative coefficients. A combinatorial explanation of ..."
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Cited by 24 (0 self)
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We consider matrices whose elements enumerate weights of walks in planar directed weighted graphs (not necessarily acyclic). These matrices are totally nonnegative; more precisely, all their minors are formal power series in edge weights with nonnegative coefficients. A combinatorial explanation of this phenomenon involves looperased walks. Applications include total positivity of hitting matrices of Brownian motion in planar domains.
Multiple qzeta values
 J. Algebra
"... Abstract. We introduce a qanalog of the multiple harmonic series commonly referred to as multiple zeta values. The multiple qzeta values satisfy a qstuffle multiplication rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values. Add ..."
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Cited by 22 (2 self)
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Abstract. We introduce a qanalog of the multiple harmonic series commonly referred to as multiple zeta values. The multiple qzeta values satisfy a qstuffle multiplication rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values. Additionally, multiple qzeta values can be viewed as special values of the multiple qpolylogarithm, which admits a multiple Jackson qintegral representation whose limiting case is the Drinfel’d simplex integral for the ordinary multiple polylogarithm when q = 1. The multiple Jackson qintegral representation for multiple qzeta values leads to a second multiplication rule satisfied by them, referred to as a qshuffle. Despite this, it appears that many numerical relations satisfied by ordinary multiple zeta values have no interesting qextension. For example, a suitable qanalog of Broadhurst’s formula for ζ({3, 1} n), if one exists, is likely to be rather complicated. Nevertheless, we show that a number of infinite classes of relations, including Hoffman’s partition identities, Ohno’s cyclic sum identities, Granville’s sum formula, Euler’s convolution formula, Ohno’s generalized duality relation, and the derivation relations of Ihara
The Algebra and Combinatorics of Shuffles and Multiple Zeta Values
, 2002
"... INTRODUCTION We continue our study of nested sums of the form i(s 1 ; s 2 ; : : : ; s k ) := n1?n2?\Delta\Delta\Delta?n k ?0 \Gammas j j ; (1) commonly referred to as multiple zeta values [2, 3, 4, 11, 12, 16, 19]. Here and throughout, s 1 ; s 2 ; : : : ; s k are positive integers with s ..."
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Cited by 19 (5 self)
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INTRODUCTION We continue our study of nested sums of the form i(s 1 ; s 2 ; : : : ; s k ) := n1?n2?\Delta\Delta\Delta?n k ?0 \Gammas j j ; (1) commonly referred to as multiple zeta values [2, 3, 4, 11, 12, 16, 19]. Here and throughout, s 1 ; s 2 ; : : : ; s k are positive integers with s 1 ? 1 to ensure convergence. 43 All rights of reproduction in any form reserved. There exist many intriguing results and conjectures concerning values of (1) at various arguments. For example, i(f3; 1g ) := i(3; 1; 3; 1; : : : ; 3; 1  z (4n + 2)! ; 0 n 2 Z; (2) was conjectured by Zagier [19] and first proved by Broadhurst et al [2] using analytic techniques. Subsequently, a purely combinatorial proof was given [3] based on the wellknown shuffle property of iterated integrals, and it is this latter approach which we develop more fully here. For further and deeper results from the analytic viewpoint, see [4]. Our main result is a generalization of (2) in which twos are
RotaBaxter algebras, dendriform algebras and PoincaréBirkhoffWitt theorem
, 2004
"... Abstract. RotaBaxter algebras appeared in both the physics and mathematics literature. It is of great interest to have a simple construction of the free object of this algebraic structure. For example, free commutative RotaBaxter algebras relate to double shuffle relations for multiple zeta values ..."
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Cited by 18 (11 self)
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Abstract. RotaBaxter algebras appeared in both the physics and mathematics literature. It is of great interest to have a simple construction of the free object of this algebraic structure. For example, free commutative RotaBaxter algebras relate to double shuffle relations for multiple zeta values. The interest in the noncommutative setting arised in connection with the work of Connes and Kreimer on the Birkhoff decomposition in renormalization theory in perturbative quantum field theory. We construct free noncommutative RotaBaxter algebras and apply the construction to obtain universal enveloping RotaBaxter algebras of dendriform dialgebras and trialgebras. We also prove an analog of the PoincaréBirkhoffWitt theorem for universal enveloping algebra in the context of dendriform trialgebras. In particular, every dendriform dialgebra and trialgebra is a subalgebra of a RotaBaxter algebra. We explicitly show that the free dendriform dialgebras and trialgebras, as represented by
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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Cited by 17 (10 self)
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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
Evaluation of Integrals Representing Correlations in XXX Heisenberg Spin Chain
 in Progress in Mathematics
, 2001
"... We study XXX Heisenberg spin 1/2 antiferromagnet. We evaluate a probability of formation of a ferromagnetic string in the antiferromagnetic ground state in thermodynamics limit. We prove that for short strings the probability can be expressed in terms of Riemann zeta function with odd arguments. ..."
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Cited by 17 (6 self)
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We study XXX Heisenberg spin 1/2 antiferromagnet. We evaluate a probability of formation of a ferromagnetic string in the antiferromagnetic ground state in thermodynamics limit. We prove that for short strings the probability can be expressed in terms of Riemann zeta function with odd arguments.
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 15 (7 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.
Renormalization of multiple zeta values
 J. Algebra
, 2006
"... Abstract. Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special ..."
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Cited by 14 (10 self)
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Abstract. Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special values of multiple zeta functions at nonpositive integers since the values are usually singular. We define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory, and following the Hopf algebra approach of Connes and Kreimer. This definition of renormalized MZVs agrees with the convergent MZVs and extends the work of IharaKanekoZagier on renormalization of MZVs with positive arguments. We further show that the important
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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Cited by 11 (0 self)
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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.