Graphical functions and single-valued multiple polylogarithms
نویسندگان
چکیده
منابع مشابه
q-Multiple Zeta Functions and q-Multiple Polylogarithms
We shall define the q-analogs 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.
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For every positive integer d we define the q-analog of multiple zeta function of depth d and study its properties, generalizing the work of Kaneko et al. who dealt with the case d = 1. We first analytically continue it to a meromorphic function on C with explicit poles. In our Main Theorem we show that its limit when q ↑ 1 is the ordinary multiple zeta function. Then we consider some special va...
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when s1, . . . , sk are positive integers and z a complex number in the unit disk. For k = 1, this is the classical polylogarithm Lis (z). These multiple polylogarithms can be defined also in terms of iterated Chen integrals and satisfy shuffle relations. Multiple polylogarithms in several variables are defined for si ≥ 1 and |zi | < 1(1 ≤ i ≤ k) by Li(s1,...,sk)(z1, . . . , zk) = ∑ n1>n2>···>n...
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Contents 1. Introduction 1 1.1. History and motivation 1 1.2. What we do 3 1.3. Plan of the article 3 1.4. Conventions and notations 4 2. The De Rham fundamental groupoid of a punctured elliptic curve 5 2.1. Nilpotent vector bundles 5 2.2. Nilpotent connections with simple poles at the origin 6 3. A universal flat connection over the upper-half plane 10 3.1. Combinatorial conventions and lemmas...
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In this talk I review the connections between Feynman integrals and multiple polylogarithms. After an introductory section on loop integrals I discuss the Mellin-Barnes transformation and shuffle algebras. In a subsequent section multiple polylogarithms are introduced. Finally, I discuss how certain Feynman integrals evaluate to multiple polylogarithms.
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ژورنال
عنوان ژورنال: Communications in Number Theory and Physics
سال: 2014
ISSN: 1931-4523,1931-4531
DOI: 10.4310/cntp.2014.v8.n4.a1