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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
padic multiple zeta values. I. padic multiple polylogarithms and the padic KZ
 equation, Invent. Math
"... Abstract. Our main aim in this paper is to give a foundation of the theory of padic multiple zeta values. We introduce (one variable) padic multiple polylogarithms by Coleman’s padic iterated integration theory. We define padic multiple zeta values to be special values of padic multiple polylog ..."
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Cited by 8 (1 self)
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Abstract. Our main aim in this paper is to give a foundation of the theory of padic multiple zeta values. We introduce (one variable) padic multiple polylogarithms by Coleman’s padic iterated integration theory. We define padic multiple zeta values to be special values of padic multiple polylogarithms. We consider the padic KZ equation and introduce the padic Drinfel’d associator by using certain two fundamental solutions of the padic KZ equation. We show that our padic multiple polylogarithms appear on coefficients of a certain fundamental solution of the padic KZ equation and our padic multiple zeta values appear on coefficients of the padic Drinfel’d associator. We show various properties of padic multiple zeta values, which are sometimes analogous to the complex case and are sometimes peculiar to the padic case, via the padic KZ equation.
The double shuffle relations for padic multiple zeta values
, 2004
"... We give a proof of double shuffle relations for padic multiple zeta values by developing higher dimensional version of tangential base points and discussing a relationship between two (and one) variable padic multiple polylogarithms and the two variable padic KZ equation. ..."
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Cited by 7 (6 self)
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We give a proof of double shuffle relations for padic multiple zeta values by developing higher dimensional version of tangential base points and discussing a relationship between two (and one) variable padic multiple polylogarithms and the two variable padic KZ equation.
PENTAGON AND HEXAGON EQUATIONS
, 2007
"... The author will prove that Drinfel’d’s pentagon equation implies his two hexagon equations. ..."
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Cited by 3 (0 self)
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The author will prove that Drinfel’d’s pentagon equation implies his two hexagon equations.
2002), A Quantum Field Theoretical Representation of EulerZagier Sums, arXiv:math.QA/9908067
 Int. J. Math. Sc. Vol
"... We establish a novel representation of arbitrary EulerZagier sums in terms of weighted vacuum graphs. This representation uses a toy quantum field theory with infinitely many propagators and interaction vertices. The propagators involve Bernoulli polynomials and Clausen functions to arbitrary order ..."
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Cited by 2 (0 self)
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We establish a novel representation of arbitrary EulerZagier sums in terms of weighted vacuum graphs. This representation uses a toy quantum field theory with infinitely many propagators and interaction vertices. The propagators involve Bernoulli polynomials and Clausen functions to arbitrary orders. The Feynman integrals of this model can be decomposed in terms of a vertex algebra whose structure we investigate. We derive a large class of relations between multiple zeta values, of arbitrary lengths and weights, using only a certain set of graphical manipulations on Feynman diagrams. Further uses and possible generalizations of Perturbative calculations of Green’s Functions in Quantum Field Theory lead to a class of iterated parameter integrals whose explicit calculation becomes very difficult beyond the first few orders in the coupling expansion
DOUBLE SHUFFLE RELATION FOR ASSOCIATORS
, 808
"... Abstract. It is proved that Drinfel’d’s pentagon equation implies the generalized double shuffle relation. As a corollary, an embedding from the GrothendieckTeichmüller group GRT1 into Racinet’s double shuffle group DMR0 is obtained, which settles the project of DeligneTerasoma. It is also proved ..."
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Cited by 1 (0 self)
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Abstract. It is proved that Drinfel’d’s pentagon equation implies the generalized double shuffle relation. As a corollary, an embedding from the GrothendieckTeichmüller group GRT1 into Racinet’s double shuffle group DMR0 is obtained, which settles the project of DeligneTerasoma. It is also proved that the gamma factorization formula follows from the generalized double shuffle relation. Contents
PERIOD POLYNOMIALS AND IHARA BRACKETS
, 2006
"... Abstract. Schneps [J. Lie Theory 16 (2006), 19–37] has found surprising links between Ihara brackets and even period polynomials. These results can be recovered and generalized by considering some identities relating Ihara brackets and classical Lie brackets. The period polynomials generated by this ..."
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Abstract. Schneps [J. Lie Theory 16 (2006), 19–37] has found surprising links between Ihara brackets and even period polynomials. These results can be recovered and generalized by considering some identities relating Ihara brackets and classical Lie brackets. The period polynomials generated by this method are found to be essentially the KohnenZagier polynomials. 1.
Contents
, 2007
"... HIDEKAZU FURUSHO Abstract. The author will prove that Drinfel’d’s pentagon equation implies ..."
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HIDEKAZU FURUSHO Abstract. The author will prove that Drinfel’d’s pentagon equation implies
Duality formulas of the Special Values of Multiple Polylogarithms
, 2003
"... The special values of multiple polylogarithms, which including multiple zeta values, appear some fields of mathematics and physics. Many kinds of their linear relations are investigated as well as their algebraic relations. From the viewpoint of a connection matrix of Fuchsian equations, we derive t ..."
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The special values of multiple polylogarithms, which including multiple zeta values, appear some fields of mathematics and physics. Many kinds of their linear relations are investigated as well as their algebraic relations. From the viewpoint of a connection matrix of Fuchsian equations, we derive two kinds of duality of these values. 1