## Extensions of the AZ-algorithm and the package multiintegrate.(English)Zbl 07458361

Bluemlein, Johannes (ed.) et al., Anti-differentiation and the calculation of Feynman amplitudes. Selected papers based on the presentations at the conference, Zeuthen, Germany, October 2020. Cham: Springer. Texts Monogr. Symb. Comput., 35-61 (2021).
This paper presents an extension of the multivariate Almkvist-Zeilberger algorithm which can compute linear differential equations for multiple integrals with one parameter. The motivation of this extension is applying the method to solve some computational problems on Feynman integrals in Quantum Field Theory. A Mathematica package MultiIntegrate was developed by the author and some concrete examples are used to show the capability of this package. With this package, the author can first try to find closed form representations of multiple hyperexponential integrals in terms of nested sums and products or iterated integrals. If a closed form is not found, the author can still succeed in computing the first coefficients of the Laurent series expansions of such integrals. In order to speed-up the computation, the author also introduced a divide-and-conquer strategy in the algorithm. The methods, algorithms, and implementations in this paper enhance the applicability of the Almkvist-Zeilberger algorithm in real applications.
For the entire collection see [Zbl 1475.81004].

### MSC:

 33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
Full Text:

### References:

 [1] J. Ablinger, Computer Algebra Algorithms for Special Functions in Particle Physics. PhD thesis, J. Kepler University Linz, 2012 [2] J. Ablinger, The package HarmonicSums: Computer algebra and analytic aspects of nested sums, in PoS LL2014, 019 (2014) [3] J. Ablinger, Inverse mellin transform of holonomic sequences, in PoS LL2016, 067 (2016) [4] J. Ablinger, Computing the inverse mellin transform of holonomic sequences using Kovacic’s algorithm, in PoS RADCOR2017, 069 (2018) [5] J. Ablinger, Discovering and proving infinite pochhammer sum identities. Experimental Mathematics (2019) [6] J. Ablinger, J. Blümlein, C. Schneider, Harmonic sums and polylogarithms generated by cyclotomic polynomials. J. Math. Phys. 52(10), 102301 (2011) · Zbl 1272.81127 [7] J. Ablinger, J. Blümlein, M. Round, C. Schneider, Advanced computer algebra algorithms for the expansion of Feynman integrals, in PoS LL2012, 050 (2012) [8] J. Ablinger, J. Blümlein, C. Schneider, Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms. J. Math. Phys. 54(8), 082301 (2013) · Zbl 1295.81071 [9] J. Ablinger, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, F. Wißbrock, The transition matrix element $$A_{}$$ gq(N) of the variable flavor number scheme at $$O(\alpha_s^3)$$. Nucl. Phys. B 882, 263-288 (2014) · Zbl 1285.81065 [10] J. Ablinger, J. Blümlein, C.G. Raab, C. Schneider, Iterated binomial sums and their associated iterated integrals. J. Math. Phys. 55(11), 112301 (2014) · Zbl 1306.81141 [11] J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, Calculating three loop ladder and V-topologies for massive operator matrix elements by computer algebra. Comput. Phys. Commun. 202, 33-112 (2016) · Zbl 1348.81034 [12] S. Abramov, M. Petkovšek, D’Alembertian solutions of linear differential and difference equations, in Proceedings of ISSAC’94, ed. by J. von zur Gathen, pp. 169-174 (1994) · Zbl 0919.34013 [13] S. Abramov, E. Zima, D’Alembertian solutions of inhomogeneous linear equations (differential, difference, and some other), in Proceedings of ISSAC’96, ed. by Y.N. Lakshman, pp. 232-240 (1996) · Zbl 0966.34005 [14] M. Apagodu, D. Zeilberger. Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory. Adv. Appl. Math. 37(2), 139-152 (2006) · Zbl 1108.05010 [15] I. Bierenbaum, J. Blümlein, S. Klein, Mellin moments of the $$o(\alpha^3_s)$$ heavy flavor contributions to unpolarized deep-inelastic scattering at $$q^2$$ ≫ $$m^2$$ and anomalous dimensions. Nucl. Phys. B 820(1), 417-482 (2009) · Zbl 1194.81244 [16] J. Blümlein, S. Kurth, Harmonic sums and Mellin transforms up to two-loop order. Phys. Rev. D 60(1), 014018 (1999) [17] J. Blümlein, M. Kauers, S. Klein, C. Schneider, Determining the closed forms of the $$O(a_s^3)$$ anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra. Comput. Phys. Commun. 180(11), 2143-2165 (2009) · Zbl 1197.81037 [18] J. Blümlein, S. Klein, C. Schneider, F. Stan, A symbolic summation approach to Feynman integral calculus. J. Symb. Comput. 47(10), 1267-1289 (2011) · Zbl 1242.81005 [19] J. Blümlein, M. Round, C. Schneider, Refined holonomic summation algorithms in particle physics. Adv. Comput. Algebra WWCA 2016 226, 51-91 (2018) · Zbl 1457.40001 [20] C. Bogner, S. Weinzierl, Feynman graph polynomials. Int. J. Modern Phys. A 25, 2585-2618 (2010) · Zbl 1193.81072 [21] D. Broadhurst, W. Zudilin, A magnetic double integral. J. Aust. Math. Soc. 1, 9-25 (2019) · Zbl 1470.11068 [22] M. Bronstein, Linear ordinary differential equations: breaking through the order 2 barrier, in Proceedings of ISSAC’92, pp. 42-48 (1992) · Zbl 0978.65507 [23] S. Chen, M. Kauers, Trading order for degree in creative telescoping. J. Symb. Comput. 47(8), 968-995 (2012) · Zbl 1241.33021 [24] S. Chen, M. Kauers, Order-degree curves for hypergeometric creative telescoping, in Proceedings of ISSAC’12, pp. 122-129 (2012) · Zbl 1323.68591 [25] S. Chen, M. Kauers, C. Koutschan, A generalized apagodu-zeilberger algorithm, in Proceedings of ISSAC’14, pp. 107-114 (2014) · Zbl 1325.68270 [26] Y. Frishman, Operator products at almost light like distances. Ann. Phys. 66, 373-389 (1971) [27] M.Y. Kalmykov, O. Veretin, Single scale diagrams and multiple binomial sums. Phys. Lett. B 483, 315-323 (2000) · Zbl 1031.81568 [28] C. Koutschan, A fast approach to creative telescoping. Math. Comput. Sci. 4(2-3), 259-266 (2010) · Zbl 1218.68205 [29] J.J. Kovacic, An algorithm for solving second order linear homogeneous differential equations. J. Symb. Comput. 2, 3-43 (1986) · Zbl 0603.68035 [30] S.-O. Moch, P. Uwer, S. Weinzierl, Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. J. Math. Phys. 43(6), 3363-3386 (2002) · Zbl 1060.33007 [31] M. Mohammed, D. Zeilberger, Sharp upper bounds for the orders of the recurrences outputted by the Zeilberger and q-Zeilberger algorithms. J. Symb. Comput. 39(2), 201-207 (2005) · Zbl 1121.33023 [32] M. Petkovšek, Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symb. Comput. 14, 243-264 (1992) · Zbl 0761.11008 [33] E. Remiddi, J.A.M. Vermaseren, Harmonic polylogarithms. Int. J. Modern Phys. A 15, 725-754 (2000) · Zbl 0951.33003 [34] C. Schneider, Symbolic summation in difference fields. PhD thesis, RISC, J. Kepler University Linz, May 2001 [35] C. Schneider, A new Sigma approach to multi-summation. Adv. Appl. Math. 34, 740-767 (2005) · Zbl 1078.33021 [36] C. Schneider, Solving parameterized linear difference equations in terms of indefinite nested sums and products. J. Differ. Equ. Appl. 11(9), 799-821 (2006) · Zbl 1087.33011 [37] C. Schneider, Symbolic summation assists combinatorics. SÃminaire Lotharingien de Combinatoire 56, 1-36 (2007) · Zbl 1188.05001 [38] C. Schneider, Simplifying multiple sums in difference fields, in Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, ed. by C. Schneider, J. Blümlein. Texts & Monographs in Symbolic Computation (Springer, Wien, 2013), pp. 325-360 · Zbl 1315.68294 [39] C. Schneider, Modern summation methods for loop integrals in quantum field theory: The packages sigma, EvaluateMultiSums and SumProduction. J. Phys. Conf. Ser. 523, 012037 (2014) [40] J.A.M. Vermaseren, Harmonic sums, Mellin transforms and integrals. Int. J. Modern Phys. A 14(13), 2037-2076 (1999) · Zbl 0939.65032 [41] K. Wegschaider, Computer generated proofs of binomial multi-sum identities. Master’s thesis, RISC, J. Kepler University, May 1997 [42] S. Weinzierl, Feynman graphs, in Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, ed. by C. Schneider, J. Blümlein. Texts & Monographs in Symbolic Computation (Springer, Wien, 2013), pp. 381-406 · Zbl 1308.81142 [43] H.S. Wilf, D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Inventiones Mathematicae 108, 575 · Zbl 0739.05007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.