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Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
, 2013
"... In recent three–loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the socalled generalized harmonic sums (in short Ssums) arise. They are characterized by rational (or real) numerator weights also different from ±1. In th ..."
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Cited by 18 (5 self)
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In recent three–loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the socalled generalized harmonic sums (in short Ssums) arise. They are characterized by rational (or real) numerator weights also different from ±1. In this article we explore the algorithmic and analytic properties of these sums systematically. We work out the Mellin and inverse Mellin transform which connects the sums under consideration with the associated Poincaré iterated integrals, also called generalized harmonic polylogarithms. In this regard, we obtain explicit analytic continuations by means of asymptotic expansions of the Ssums which started to occur frequently in current QCD calculations. In addition, we derive algebraic and structural relations, like differentiation w.r.t. the external summation index and different multiargument relations, for the compactification of Ssum expressions. Finally, we calculate algebraic relations for infinite Ssums, or equivalently for generalized harmonic polylogarithms evaluated at special values. The corresponding algorithms and relations are encoded in the computer algebra package HarmonicSums.
Fast Algorithms for Refined Parameterized Telescoping in Difference Fields
 in : Lecture Notes in Computer Science
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Simplifying multiple sums in difference fields
 in Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions
, 2013
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Solving Linear Recurrence Equations With Polynomial Coefficients
"... Summation is closely related to solving linear recurrence equations, since an indefinite sum satisfies a firstorder linear recurrence with constant coefficients, and a definite properhypergeometric sum satisfies a linear recurrence with polynomial coefficients. Conversely, d’Alembertian solutions ..."
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Cited by 2 (0 self)
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Summation is closely related to solving linear recurrence equations, since an indefinite sum satisfies a firstorder linear recurrence with constant coefficients, and a definite properhypergeometric sum satisfies a linear recurrence with polynomial coefficients. Conversely, d’Alembertian solutions of linear recurrences can be expressed as nested indefinite sums with hypergeometric summands. We sketch the simplest algorithms for finding polynomial, rational, hypergeometric, d’Alembertian, and Liouvillian solutions of linear recurrences with polynomial coefficients, and refer to the relevant literature for stateoftheart algorithms for these tasks. We outline an algorithm for finding the minimal annihilator of a given Precursive sequence, prove the salient closure properties of d’Alembertian sequences, and present an alternative proof of a recent result of Reutenauer’s that Liouvillian sequences are precisely the interlacings of d’Alembertian ones.
Nested (inverse) binomial sums and new iterated integrals for massive Feynman diagrams
, 2014
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P o S(LL2014)052 Nonplanar Feynman integrals, MellinBarnes
, 2014
"... The construction of MellinBarnes (MB) representations for nonplanar Feynman diagrams and the summation of multiple series derived from general MB representations are discussed. A basic version of a new package AMBREv.3.0 is supplemented. The ultimate goal of this project is the automatic evaluatio ..."
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The construction of MellinBarnes (MB) representations for nonplanar Feynman diagrams and the summation of multiple series derived from general MB representations are discussed. A basic version of a new package AMBREv.3.0 is supplemented. The ultimate goal of this project is the automatic evaluation of MB representations for multiloop scalar and tensor Feynman integrals through infinite sums, preferably with analytic solutions. We shortly describe a strategy of further algebraic summation.
Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations
 P OS(LL2014)017
, 2014
"... We outline a new algorithm to solve coupled systems of differential equations in one continuous variable x (resp. coupled difference equations in one discrete variable N) depending on a small parameter ε: given such a system and given sufficiently many initial values, we can determine the first coef ..."
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We outline a new algorithm to solve coupled systems of differential equations in one continuous variable x (resp. coupled difference equations in one discrete variable N) depending on a small parameter ε: given such a system and given sufficiently many initial values, we can determine the first coefficients of the Laurentseries solutions in ε if they are expressible in terms of indefinite nested sums and products. This systematic approach is based on symbolic summation algorithms in the context of difference rings/fields and uncoupling algorithms. The proposed method gives rise to new interesting applications in connection with integration by parts (IBP) methods. As an illustrative example, we will demonstrate how one can calculate the εexpansion of a ladder graph with 6 massive fermion lines.