A Generalization of the Duality and Sum Formulas on the Multiple Zeta Values

Ohno, Yasuo

doi:10.1006/jnth.1998.2314

Cited by 158 publications

(144 citation statements)

References 7 publications

(5 reference statements)

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“…which was originally conjectured independently by C. Moen and M. Schmidt around 1990 [11,12,13]. Also he mentioned that it was proved independently by Zagier in one of his unpublished papers.…”

Section: Shuffle Relations and The Sum Formulamentioning

confidence: 82%

“…When p = 1, or ( p, q) = (2, 4), or ( p, q) = (4, 2), or p = q, or p + q is odd, S p,q can be expressed in terms of the special values of Riemann zeta function at positive integers. See [12,13,14] for the details of evaluations.…”

mentioning

confidence: 99%

“…Multiple zeta values are multidimensional version of the Euler sums [1,6,9,13,14,15]. For a string of positive integers α = (α 1 , α 2 , .…”

mentioning

confidence: 99%

“…There is an integral representation, due to Kontsevich [4,5,13], to express multiple zeta values in terms of iterated integrals (or Drinfeld integrals) over simplices of weight-dimension, namely, MINKING EIE, FU-YAO YANG and YAO LIN ONG 377 and j = dt j /t j , otherwise. For our convenience, we rewrite the above integral representation as…”

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confidence: 99%

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Some alternating double sum formulae of multiple zeta values

Eie¹,

Yang²,

Ong³

2010

Comput. Appl. Math.

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Abstract. In this paper, we produce shuffle relations from multiple zeta values of the form ζ ({1} m−1 , n + 1). Here {1} k is k repetitions of 1, and for a string of positive integers α 1 , α 2 , . . . , α r with α r ≥ 2As applications of the sum formula and a newly developed weighted sum formula, we shall prove for even integers k, r ≥ 0 thatMathematical subject classification: Primary: 40A25, 40B05; Secondary: 11M99, 33E99.

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Section: Shuffle Relations and The Sum Formulamentioning

confidence: 82%

mentioning

confidence: 99%

“…Multiple zeta values are multidimensional version of the Euler sums [1,6,9,13,14,15]. For a string of positive integers α = (α 1 , α 2 , .…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Some alternating double sum formulae of multiple zeta values

Eie¹,

Yang²,

Ong³

2010

Comput. Appl. Math.

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“…Indeed, many relations among multiple zeta values are obtained by studying relations among MZV fractions. The method, the so called partial fractional method, can be traced back to Euler in the case when k = 2 and remains one of the most effective methods until today [5,8,15].…”

Section: Introductionmentioning

confidence: 99%

The shuffle relation of fractions from multiple zeta values

Guo

Xie

2011

Ramanujan J

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Abstract. Partial fraction methods play an important role in the study of multiple zeta values. One class of such fractions is related to the integral representations of MZVs. We show that this class of fractions has a natural structure of shuffle algebra. This finding conceptualizes the connections among the various methods of stuffle, shuffle and partial fractions in the study of MZVs. This approach also gives an explicit product formula of the fractions.

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