[bib]

@**incollection**{RISC6408,

author = {J. Ablinger},

title = {{Extensions of the AZ-algorithm and the Package MultiIntegrate}},

booktitle = {{Anti-Differentiation and the Calculation of Feynman Amplitudes}},

language = {english},

abstract = {We extend the (continuous) multivariate Almkvist-Zeilberger algorithm inorder to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals overhyperexponential integrals and try to find closed form representations in terms ofnested sums and products or iterated integrals. In addition, if we fail to computea closed form solution in full generality, we may succeed in computing the firstcoeffcients of the Laurent series expansions of such integrals in terms of indefnitenested 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) multivariateAlmkvist Zeilberger algorithm to compute recurrences or differential equations forhyperexponential integrands and integrals. Together with the summation packageSigma and the package HarmonicSums our package provides methods to computeclosed form representations (or coeffcients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums oriterated integrals.},

series = {Texts & Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria)},

pages = {35--61},

publisher = {Springer},

isbn_issn = {ISBN 978-3-030-80218-9},

year = {2021},

note = {arXiv:2101.11385 [cs.SC]},

editor = {J. Blümlein and C. Schneider},

refereed = {yes},

keywords = {multivariate Almkvist-Zeilberger algorithm, hyperexponential integrals, iterated integrals, nested sums},

length = {27},

url = {https://doi.org/10.1007/978-3-030-80219-6_2}

}