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Contiguous Relations and Creative Telescoping

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Anti-Differentiation and the Calculation of Feynman Amplitudes

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Abstract

This article presents an algorithmic theory of contiguous relations. Contiguous relations, first studied by Gauß, are a fundamental concept within the theory of hypergeometric series. In contrast to Takayama’s approach, which for elimination uses non-commutative Gröbner bases, our framework is based on parameterized telescoping and can be viewed as an extension of Zeilberger’s creative telescoping paradigm based on Gosper’s algorithm. The wide range of applications include elementary algorithmic explanations of the existence of classical formulas for non-terminating hypergeometric series such as Gauß, Pfaff-Saalschütz, or Dixon summation. The method can be used to derive new theorems, like a non-terminating extension of a classical recurrence established by Wilson between terminating 4 F 3-series. Moreover, our setting helps to explain the non-minimal order phenomenon of Zeilberger’s algorithm.

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Notes

  1. 1.

    The number of these relations can be reduced further by taking symmetries (e.g., swapping a and b) into account.

  2. 2.

    We are using the Mathematica package “fastZeil” presented in [22]; it is freely available as described at https://combinatorics.risc.jku.at/software.

  3. 3.

    The conditions on the \(b_j-\beta _j^{(l)}\) remain valid as being those for p F q bottom parameters.

  4. 4.

    Recall Definition 2.

  5. 5.

    Notice that the right-hand sum is obtained by replacing a 1 with a 1 − 1 in the left sum.

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Acknowledgements

Part of this work has been carried out at the Department of Mathematics of the Pennsylvania State University. I wish to thank George Andrews for his overwhelming hospitality and for providing a wonderful working environment. For valuable feed-back on an early version of this article I want to acknowledge my thanks to George Andrews, Richard Askey, Frédéric Chyzak, Tom Koornwinder, Christian Krattenthaler, Bruno Salvy, Raimundas Vidunas, Herb Wilf, and Doron Zeilberger. Special thanks go to the anonymous referee who read the manuscript extremely carefully and whose suggestions for improvement were highly appreciated.

Partially supported by SFB-grant F50-06 of the Austrian Science Fund FWF, and by a visiting researcher grant of the Department of Mathematics of the Pennsylvania State University.

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Paule, P. (2021). Contiguous Relations and Creative Telescoping. In: Blümlein, J., Schneider, C. (eds) Anti-Differentiation and the Calculation of Feynman Amplitudes. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-80219-6_15

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