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The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré– iterated integrals including denominators of higher cyclotomic polynomials. We derive the… (More)

In recent three–loop calculations of massive Feynman integrals within Quantum Chromody-namics (QCD) and, e.g., in recent combinatorial problems the so-called generalized harmonic sums (in short S-sums) arise. They are characterized by rational (or real) numerator weights also different from ±1. In this article we explore the algorithmic and analytic… (More)

In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and massive Feynman diagrams at higher loops. These quantities are elements of stuffle and shuffle algebras implying algebraic relations being widely independent of… (More)

Two-point Feynman parameter integrals, with at most one mass and containing local operator insertions in 4 + ε-dimensional Minkowski space, can be transformed to multi-integrals or multi-sums over hyperexponential and/or hypergeometric functions depending on a discrete parameter n. Given such a specific representation, we utilize an enhanced version of the… (More)

Three loop ladder and V-topology diagrams contributing to the massive operator matrix element A Qg are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable N and the dimensional parameter ε. Given these representations, the desired Laurent series expansions in ε can be obtained… (More)

- Jakob Ablinger, Jakob Ablinger Kurzfassung, Ii Kurzfassung
- 2009

Ich erkläre hiermit an Eides statt, dass ich die vorliegende Arbeit selbstständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen nicht benützt und die den benutzten Quellen wörtlich oder inhaltlich entnommenen Stellen als solche kenntlich gemacht habe. Abstract III Abstract In this work we present the computer algebra package… (More)

- Jakob Ablinger
- 2014

This paper summarizes the essential functionality of the computer algebra package HarmonicSums. On the one hand HarmonicSums can work with nested sums such as harmonic sums and their generalizations and on the other hand it can treat iterated integrals of the Poincaré and Chen-type, such as harmonic polylogarithms and their generalizations. The interplay of… (More)

An expression in terms of (cyclotomic) harmonic sums can be simplified by the quasi-shuffle algebra in terms of the so-called basis sums. By construction, these sums are algebraically independent within the quasi-shuffle algebra. In this article we show that the basis sums can be represented within a tower of difference ring extensions where the constants… (More)

We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do not request special choices of bases. Here we assume that the desired solution has a power series representation and we seek for the coefficients in closed form. In… (More)