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000598863 245__ $$aHypergeometric Structures in Feynman Integrals
000598863 260__ $$aDordrecht [u.a.]$$bSpringer Science + Business Media B.V$$c2023
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000598863 520__ $$aHypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {\tt Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {\tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {\tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.
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000598863 650_7 $$2INSPIRE$$aFeynman graph
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000598863 650_7 $$2INSPIRE$$amathematical methods
000598863 650_7 $$2INSPIRE$$aTaylor expansion
000598863 650_7 $$2INSPIRE$$atopology
000598863 650_7 $$2INSPIRE$$amaster integral
000598863 650_7 $$2INSPIRE$$acomputer: algebra
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000598863 7001_ $$0P:(DE-H253)PIP1086086$$aSaragnese, Marco$$b1
000598863 7001_ $$0P:(DE-H253)PIP1093315$$aSchneider, Carsten$$b2$$eCorresponding author
000598863 773__ $$0PERI:(DE-600)2002961-5$$a10.1007/s10472-023-09831-8$$gVol. 91, no. 5, p. 591 - 649$$n5$$p591 - 649$$tAnnals of mathematics and artificial intelligence$$v91$$x1012-2443$$y2023
000598863 7870_ $$0PUBDB-2021-04789$$aBlümlein, Johannes et.al.$$d2021$$iIsParent$$rDESY-21-071 ; arXiv:2111.15501 ; DO-TH-21-16 ; RISC Report Series 21-17 ; SAGEX-21-10-E$$tHypergeometric Structures in Feynman Integrals
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