Simplifying multiple sums in difference fields. (English) Zbl 1315.68294

Schneider, Carsten

Schneider, Carsten (ed.) et al., Computer algebra in quantum field theory. Integration, summation and special functions. Wien: Springer (ISBN 978-3-7091-1615-9/hbk; 978-1-4614-8523-0/ebook). Texts and Monographs in Symbolic Computation, 325-360 (2013).

Summary: 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 Sigma by discovering and proving new harmonic number identities extending those from [P. Paule and C. Schneider, Adv. Appl. Math. 31, No. 2, 359–378 (2003; Zbl 1039.11007)]. In addition, the newly developed package 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.
For the entire collection see [Zbl 1276.81004].

MSC:

68W30	Symbolic computation and algebraic computation
12H10	Difference algebra
12Y05	Computational aspects of field theory and polynomials (MSC2010)
81T18	Feynman diagrams

Keywords:

survey article; difference field algorithms; symbolic summation; large-scale summation problems; Feynman diagrams; quantum chromodynamics

Software:

Dependencies; EvaluateMultiSums; Mathematica; SIGMA