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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 40 (30 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
qmultiple zeta functions and qmultiple polylogarithms. preprint, 2003, arXiv:math.QA/0304448
"... Abstract. We shall define the qanalogs of multiple zeta functions and multiple polylogarithms in this paper and study their properties, based on the work of Kaneko et al. and Schlesinger, respectively. 1 Introduction and definitions Let 0 < q < 1 and for any positive integer k define its qan ..."
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Cited by 13 (1 self)
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Abstract. We shall define the qanalogs of multiple zeta functions and multiple polylogarithms in this paper and study their properties, based on the work of Kaneko et al. and Schlesinger, respectively. 1 Introduction and definitions Let 0 < q < 1 and for any positive integer k define its qanalog [k] = [k]q = (1 − qn)/(1 − q). In [5] Kaneko et al. define a function of two complex variables fq(s; t) = ∑∞ k=1 qkt /[k] s such that the qanalog of Riemann zeta function is realized as
An integral representation of multiple HurwitzLerch zeta functions and generalized multiple bernoulli
, 2009
"... A surface integral representation of a multiple generalization of the Hurwitz–Lerch zeta function is given, which is a direct analogue of the wellknown contour integral representation of the Riemann zeta function of Hankel’s type. From this integral representation, we derive a detailed description ..."
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Cited by 10 (5 self)
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A surface integral representation of a multiple generalization of the Hurwitz–Lerch zeta function is given, which is a direct analogue of the wellknown contour integral representation of the Riemann zeta function of Hankel’s type. From this integral representation, we derive a detailed description of the set of its possible singularities. In addition, we present two formulae for special values of the zeta function at nonpositive integers in terms of generalizations of Bernoulli numbers. These results are refinements of previously known ones. 1.
DIFFERENTIAL ALGEBRAIC BIRKHOFF DECOMPOSITION AND THE RENORMALIZATION OF MULTIPLE ZETA VALUES
"... Abstract. In the Hopf algebra approach of Connes and Kreimer on renormalization of quantum field theory, the renormalization process is viewed as a special case of the Algebraic Birkhoff Decomposition. We give a differential algebra variation of this decomposition and apply this to the study of mult ..."
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Cited by 9 (6 self)
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Abstract. In the Hopf algebra approach of Connes and Kreimer on renormalization of quantum field theory, the renormalization process is viewed as a special case of the Algebraic Birkhoff Decomposition. We give a differential algebra variation of this decomposition and apply this to the study of multiple zeta values. 1.
DOUBLE ZETA VALUES AND MODULAR FORMS
"... Dedicated to the memory of Tsuneo Arakawa ..."
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Evaluations of multiple Dirichlet Lvalues via symmetric functions
 J. Number Theory
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The AkiyamaTanigawa algorithm for Carlitz’s qBernoulli numbers
 INTEGERS: ELECTRONIC J. COMBIN. NUMBER THEORY 6
, 2006
"... We show that the AkiyamaTanigawa algorithm and Chen’s variant for computing Bernoulli numbers can be generalized to Carlitz’s qBernoulli numbers. We also put these algorithms in the larger context of generalized EulerSeidel matrices. ..."
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Cited by 4 (2 self)
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We show that the AkiyamaTanigawa algorithm and Chen’s variant for computing Bernoulli numbers can be generalized to Carlitz’s qBernoulli numbers. We also put these algorithms in the larger context of generalized EulerSeidel matrices.
Algorithms for Bernoulli and allied polynomials
 J. Integer Seq
"... We investigate some algorithms that produce Bernoulli, Euler and Genocchi polynomials. We also give closed formulas for Bernoulli, Euler and Genocchi polynomials in terms of weighted Stirling numbers of the second kind, which are extensions of known formulas for Bernoulli, Euler and Genocchi numbers ..."
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Cited by 4 (2 self)
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We investigate some algorithms that produce Bernoulli, Euler and Genocchi polynomials. We also give closed formulas for Bernoulli, Euler and Genocchi polynomials in terms of weighted Stirling numbers of the second kind, which are extensions of known formulas for Bernoulli, Euler and Genocchi numbers involving Stirling numbers of the second kind. 1
Differential Birkhoff decomposition and the renormalization of multiple zeta values
 J. Number Theory
, 2007
"... Abstract. In the Hopf algebra approach of Connes and Kreimer on renormalization of quantum field theory, the renormalization process is views as a special case of the Algebraic Birkhoff Decomposition. We give a differential algebra variation of this decomposition and apply this to the study of multi ..."
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Cited by 3 (1 self)
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Abstract. In the Hopf algebra approach of Connes and Kreimer on renormalization of quantum field theory, the renormalization process is views as a special case of the Algebraic Birkhoff Decomposition. We give a differential algebra variation of this decomposition and apply this to the study of multiple zeta values. 1.
Desingularization of complex multiple zetafunctions, fundamentals of padic multiple Lfunctions, and evaluation of their special values
"... Abstract. This paper deals with a multiple version of zeta and Lfunctions both in the complex case and in the padic case: (I). Our motivation in the complex case is to find suitable rigorous meaning of the values of multivariable multiple zetafunctions at nonpositive integer points. (a). A desi ..."
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Abstract. This paper deals with a multiple version of zeta and Lfunctions both in the complex case and in the padic case: (I). Our motivation in the complex case is to find suitable rigorous meaning of the values of multivariable multiple zetafunctions at nonpositive integer points. (a). A desingularization of multiple zetafunctions (of the generalized EulerZagier type): We reveal that multiple zetafunctions (which are known to be meromorphic in the whole space with whose singularities lying on infinitely many hyperplanes) turn to be entire on the whole space after taking the desingularization. Further we show that the desingularized function is given by a suitable finite ‘linear ’ combination of multiple zetafunctions with some arguments shifted. It is also shown that specific combinations of Bernoulli numbers attain the special values at their nonpositive integers of the desingularized ones. (b). Twisted multiple zetafunctions: Those can be continued to entire functions, and their special values at nonpositive integer points can be explicitly calculated. (II). Our work in the padic case is to develop the study on analytic side of the KubotaLeopoldt padic Lfunctions into the multiple setting. We construct padic multiple L