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 Sigma by discovering and proving new harmonic number identities extending those from Paule and Schneider, 2003. 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.

References in zbMATH (referenced in 14 articles , 1 standard article )

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  1. Behring, A.; Blümlein, J.; De Freitas, A.; Goedicke, A.; Klein, S.; von Manteuffel, A.; Schneider, C.; Schönwald, K.: The polarized three-loop anomalous dimensions from on-shell massive operator matrix elements (2019)
  2. Stenlund, David; Wan, James G.: Some double sums involving ratios of binomial coefficients arising from urn models (2019)
  3. Ablinger, Jakob; Schneider, Carsten: Algebraic independence of sequences generated by (cyclotomic) harmonic sums (2018)
  4. de Hon, Bastiaan P.; Floris, Sander J.; Arnold, John M.: No-neighbours recurrence schemes for space-time Green’s functions on a 3D simple cubic lattice (2018)
  5. Ablinger, J.; Behring, A.; Blümlein, J.; De Freitas, A.; von Manteuffel, A.; Schneider, C.: The three-loop splitting functions (P_q g^(2)) and (P_g g^(2, \operatornameN_\operatornameF)) (2017)
  6. Chen, Shaoshi; Kauers, Manuel: Some open problems related to creative telescoping (2017)
  7. Schneider, Carsten: Summation theory. II: Characterizations of (R \Pi\Sigma^\ast)-extensions and algorithmic aspects (2017)
  8. Schneider, Carsten; Sulzgruber, Robin: Asymptotic and exact results on the complexity of the Novelli-Pak-Stoyanovskii algorithm (2017)
  9. Ablinger, J.; Behring, A.; Blümlein, J.; De Freitas, A.; von Manteuffel, A.; Schneider, C.: Calculating three loop ladder and (V)-topologies for massive operator matrix elements by computer algebra (2016)
  10. Schneider, Carsten: A difference ring theory for symbolic summation (2016)
  11. Kauers, Manuel; Yen, Lily: On the length of integers in telescopers for proper hypergeometric terms (2015)
  12. Ablinger, J.; Blümlein, J.; Raab, C. G.; Schneider, C.: Iterated binomial sums and their associated iterated integrals (2014)
  13. Ulmer, Douglas: Curves and Jacobians over function fields (2014)
  14. Schneider, Carsten: Simplifying multiple sums in difference fields (2013)