Abstract
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.
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Notes
- 1.
- 2.
\(\mathbb{Z}\) are the integers, \(\mathbb{N} =\{ 0, 1, 2,\ldots \}\) are the non-negative integers, and all fields (resp. rings) contain the rational numbers \(\mathbb{Q}\) as a subfield (resp. subring). For a set A we define A ∗:= A∖{0}.
- 3.
The solution set \(V =\{ (c_{0},\ldots,c_{d},g) \in {\mathbb{K}}^{d+1} \times \mathbb{F}\vert \alpha _{1}\sigma (g) + \alpha _{0}g =\sum _{ i=0}^{d}c_{i}f_{i}\}\) forms a \(\mathbb{K}\)-vector space of dimension ≤ d + 2 and the task is to get an explicit basis of V.
- 4.
The solution set \(V =\{ (c_{0},\ldots,c_{d},g) \in {\mathbb{K}}^{d+1} \times \mathbb{F}\;\vert \; \text{ holds}\}\) forms a \(\mathbb{K}\)-vector space of dimension ≤ m + d + 1 and the task is to get an explicit basis of V.
- 5.
\(\mathbb{K}(x)[p_{1},p_{1}^{-1},\ldots,p_{r},p_{r}^{-1}]\) stands for the polynomial Laurent ring in the variables \(p_{1},\ldots,p_{r}\), i.e., an element is of the form \(\sum _{(i_{1},\ldots,i_{r})\in S}f_{(i_{1},\ldots,i_{r})}p_{1}^{i_{1}}\ldots p_{r}^{i_{r}}\) where \(f_{(i_{1},\ldots,i_{r})} \in \mathbb{K}(x)\) and \(S \subseteq {\mathbb{Z}}^{r}\) is finite.
- 6.
As observed in Remark 2 one might need in addition the alternating sign to represent all hypergeometric products. The underlying solution works analogously by adapted algorithms.
- 7.
\(L(m_{1},\ldots,m_{u},n)\), u ≥ 0, stands for a linear combination of the m i and n with integer coefficients.
- 8.
For a rigorous verification the proof certificate \((c_{0},\ldots,c_{d},G(k))\) of (41) with d = 2 is returned with the function call CreativeTelescoping[mySum,n].
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Supported by the Austrian Science Fund (FWF) grants P20347-N18 and SFB F50 (F5009-N15) and by the EU Network LHCPhenoNet PITN-GA-2010-264564.
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Schneider, C. (2013). Simplifying Multiple Sums in Difference Fields. In: Schneider, C., Blümlein, J. (eds) Computer Algebra in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1616-6_14
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