## Abstract

We extend the (continuous) multivariate Almkvist-Zeilberger algorithm in order to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals over hyperexponential integrals and try to find closed form representations in terms of nested sums and products or iterated integrals. In addition, if we fail to compute a closed form solution in full generality, we may succeed in computing the first coefficients of the Laurent series expansions of such integrals in terms of indefinite nested sums and products or iterated integrals. In this article we present the corresponding methods and algorithms. Our Mathematica package MultiIntegrate, can be considered as an enhanced implementation of the (continuous) multivariate Almkvist Zeilberger algorithm to compute recurrences or differential equations for hyperexponential integrands and integrals. Together with the summation package Sigma and the package HarmonicSums our package provides methods to compute closed form representations (or coefficients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums or iterated integrals.

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## Notes

- 1.
The Mathematica packages MultiIntegrate, HarmonicSums, Sigma and EvaluateMultiSums can be downloaded at https://risc.jku.at/software.

- 2.
There exist upper bounds for a particular input. But usually, these bounds are too high and one tries smaller values.

- 3.
There exist upper bounds for a particular input. But usually, these bounds are too high and one tries smaller values.

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## Acknowledgements

This work was supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15) and by the bilateral project WTZ BG 03/2019 (KP-06-Austria/8/2019), funded by OeAD (Austria) and Bulgarian National Science Fund. The author would like to thank C. Schneider for useful discussions.

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Ablinger, J. (2021). Extensions of the AZ-Algorithm and the Package MultiIntegrate. 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_2

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