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Refined Holonomic Summation Algorithms in Particle Physics

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Advances in Computer Algebra (WWCA 2016)

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Abstract

An improved multi-summation approach is introduced and discussed that enables one to simultaneously handle sequences generated by indefinite nested sums and products in the setting of difference rings and holonomic sequences described by linear recurrence systems. Relevant mathematics is reviewed and the underlying advanced difference ring machinery is elaborated upon. The flexibility of this new toolbox contributed substantially to evaluating complicated multi-sums coming from particle physics. Illustrative examples of the functionality of the new software package RhoSum are given.

Dedicated to Sergei A. Abramov on the occasion of his 70th birthday

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Notes

  1. 1.

    \(L(n_1,\dots ,n_l)\) stands for \(z_0+z_1\,n_1+\dots +z_l\,n_l\) for some integers \(z_0,\dots ,z_l\).

  2. 2.

    For a finite set \(L\subseteq \mathbb {A}\) we define \(\text {supp}(L)=\{1\leqslant j\leqslant e\mid t_j\text { occurs in }L\}\).

  3. 3.

    In order to apply our summation algorithms, we must assume that the constant field \(\mathbb {K}=\text {const}_{\sigma }\mathbb {G}\) has certain algorithmic properties [55]; this is guaranteed if we are given, e.g., a rational function field \(\mathbb {K}=\mathbb {K}'(x_1,\dots ,x_l)\) over an algebraic number field \(\mathbb {K}'\).

  4. 4.

    If one is only interested in the telescoping problem with \(d=1\), it might be worthwhile to look for a solution of (14) in a \(\mathbb {G}\)-simple \(R\varPi \varSigma \)-extension; this particular case is neglected in the following.

  5. 5.

    For the rational and q-rational difference fields see also [15, 16, 39].

  6. 6.

    We will make this statement precise in Theorem 2.3 by choosing specific variants of Problem RPT.

  7. 7.

    Also in  [34] coupled systems are constructed to handle multi-sums. Here we restrict to a special form so that the full power of our tools from Sect. 2 can be applied without using any Gröber bases or uncoupling computations. In particular, the recurrences can have inhomogeneous parts which can be represented in \(\varPi \varSigma \)-fields and \(R\varPi \varSigma \)-extensions. Also the coefficients could be represented in general \(\varPi \varSigma \)-fields (see [54]), but we will skip this more exotic case.

  8. 8.

    If there are exceptional points within the summation range, we refer to Sect. 3.2.3. Further, if the upper bound is \(\infty \), limit computations are necessary. For wide classes of indefinite nested sums asymptotic expansions can be computed [1, 10, 12, 13] that can be used for this task.

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Acknowledgements

This work was supported in part by the guest programme of Kolleg Mathematik-Physik Berlin (KMPB), the European Commission through contract PITN-GA-2012-316704 (HIGGSTOOLS), and by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15).

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  1. Deutsches Elektronen-Synchroton (DESY), Zeuthen, Platanenalle 6, 15738, Zeuthen, Germany

    Johannes Blümlein

  2. Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenberger Straße 69, 4040, Linz, Austria

    Mark Round & Carsten Schneider

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  1. Johannes Blümlein
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  1. Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria

    Carsten Schneider

  2. Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Eugene Zima

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Blümlein, J., Round, M., Schneider, C. (2018). Refined Holonomic Summation Algorithms in Particle Physics. In: Schneider, C., Zima, E. (eds) Advances in Computer Algebra. WWCA 2016. Springer Proceedings in Mathematics & Statistics, vol 226. Springer, Cham. https://doi.org/10.1007/978-3-319-73232-9_3

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