Nested (inverse) binomial sums and new iterated integrals for massive Feynman diagrams

Nested sums containing binomial coefficients occur in the computation of massive operator matrix elements. Their associated iterated integrals lead to alphabets including radicals, for which we determined a suitable basis. We discuss algorithms for converting between sum and integral representations, mainly relying on the Mellin transform. To aid the conversion we worked out dedicated rewrite rules, based on which also some general patterns emerging in the process can be obtained.Comment: 13 pages LATEX, one style file, Proceedings of Loops and Legs in Quantum Field Theory -- LL2014,27 April 2014 -- 02 May 2014 Weimar, German

Non-planar Feynman integrals, Mellin-Barnes representations, multiple sums

The construction of Mellin-Barnes (MB) representations for non-planar 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.Comment: Contribution to the proceedings of the Loops and Legs 2014 conferenc

New Results on Massive 3-Loop Wilson Coefficients in Deep-Inelastic Scattering

We present recent results on newly calculated 2- and 3-loop contributions to the heavy quark parts of the structure functions in deep-inelastic scattering due to charm and bottom.Comment: Contribution to the Proc. of Loops and Legs 2016, PoS, in prin

Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations

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 $\epsilon$: given such a system and given sufficiently many initial values, we can determine the first coefficients of the Laurent-series solutions in $\epsilon$ 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 $\epsilon$-expansion of a ladder graph with 6 massive fermion lines

Simplifying Multiple Sums in Difference Fields

In this survey article we present difference field algorithms for symbolic summation. Special emphasize is put on new aspects in how the summation problems are rephrased in terms of difference fields, how the problems are solved there, and how the derived results in the given difference field can be reinterpreted as solutions of the input problem. The algorithms are illustrated with the Mathematica package \SigmaP\ by discovering and proving new harmonic number identities extending those from (Paule and Schneider, 2003). In addition, the newly developed package \texttt{EvaluateMultiSums} is introduced that combines the presented tools. In this way, large scale summation problems for the evaluation of Feynman diagrams in QCD (Quantum ChromoDynamics) can be solved completely automatically.Comment: Uses svmult.cls, to appear as contribution in the book "Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions" (www.Springer.com