scholarly journals Multiple Polylogarithms: An Introduction

2002 ◽  
pp. 1-12 ◽  
Author(s):  
M. Waldschmidt
Keyword(s):  
Multiple Polylogarithms

scholarly journals Clean Single-Valued Polylogarithms

2021 ◽  
Author(s):  
Steven Charlton ◽  
◽  
Claude Duhr ◽  
Herbert Gangl ◽  
◽  
...  
Keyword(s):  
Weight Functions ◽  
Explicit Formulas ◽  
Basic Properties ◽  
Lower Weight ◽  
Functional Relations ◽  
Multiple Polylogarithms ◽  
Real Analytic

We define a variant of real-analytic polylogarithms that are single-valued and that satisfy ''clean'' functional relations that do not involve any products of lower weight functions. We discuss the basic properties of these functions and, for depths one and two, we present some explicit formulas and results. We also give explicit formulas for the single-valued and clean single-valued version attached to the Nielsen polylogarithms Sn,2(x), and we show how the clean single-valued functions give new evaluations of multiple polylogarithms at certain algebraic points.

scholarly journals Symbolic integration and multiple polylogarithms

2013 ◽  
Author(s):  
Christian Bogner ◽  
Francis Brown
Keyword(s):  
Symbolic Integration ◽  
Multiple Polylogarithms

scholarly journals Generalized multiple Dirichlet series and generalized multiple polylogarithms

2006 ◽  
Vol 124 (2) ◽  
pp. 139-158 ◽  
Author(s):  
Kohji Matsumoto ◽  
Hirofumi Tsumura
Keyword(s):  
Dirichlet Series ◽  
Multiple Polylogarithms ◽  
Multiple Dirichlet Series

scholarly journals The diagrammatic coaction beyond one loop

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Samuel Abreu ◽  
Ruth Britto ◽  
Claude Duhr ◽  
Einan Gardi ◽  
James Matthew
Keyword(s):  
Differential Equations ◽  
Hypergeometric Functions ◽  
Differential Forms ◽  
Feynman Graph ◽  
Loop Order ◽  
Feynman Integrals ◽  
Mathematical Properties ◽  
Multiple Polylogarithms ◽  
Complete Set ◽  
General Tool

Abstract The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the ϵ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties.

scholarly journals Skewed Sudakov regime, harmonic numbers, and multiple polylogarithms

2019 ◽  
Vol 99 (2) ◽  
Author(s):  
Victor T. Kim ◽  
Victor A. Matveev ◽  
Grigorii B. Pivovarov
Keyword(s):  
Harmonic Numbers ◽  
Multiple Polylogarithms

scholarly journals Iterated integrals on products of one variable multiple polylogarithms

2020 ◽  
Vol 16 (10) ◽  
pp. 2167-2186
Author(s):  
Jiangtao Li
Keyword(s):  
Multiple Zeta Values ◽  
Iterated Integrals ◽  
Multiple Zeta ◽  
Multiple Polylogarithms ◽  
Series Representations ◽  
Zeta Values

In this paper, we show that the iterated integrals on products of one variable multiple polylogarithms from [Formula: see text] to [Formula: see text] are actually in the algebra of multiple zeta values if they are convergent. In the divergent case, we define the regularized iterated integrals from [Formula: see text] to [Formula: see text]. By the same method, we show that the regularized iterated integrals are also in the algebra of multiple zeta values. As an application, we give new series representations for multiple zeta values and calculate some interesting examples of iterated integrals.

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