Bytev, Vladimir V.; Kalmykov, Mikhail Yu.; Kniehl, Bernd A. Differential reduction of generalized hypergeometric functions from Feynman diagrams: one-variable case. (English) Zbl 1206.81089 Nucl. Phys., B 836, No. 3, 129-170 (2010). Summary: The differential-reduction algorithm, which allows one to express generalized hypergeometric functions with parameters of arbitrary values in terms of the same functions with parameters whose values differ from the original ones by integers, is discussed in the context of evaluating Feynman diagrams. Where this is possible, we compare our results with those obtained using standard techniques. It is shown that the criterion of reducibility of multiloop Feynman integrals can be reformulated in terms of the criterion of reducibility of hypergeometric functions. The relation between the numbers of master integrals obtained by differential reduction and integration by parts is discussed. Cited in 16 Documents MSC: 81T18 Feynman diagrams 33C20 Generalized hypergeometric series, \({}_pF_q\) 30K05 Universal Taylor series in one complex variable Keywords:generalized hypergeometric functions; differential reduction; Laurent expansion; multiloop calculations Software:Nestedsums; AMBRE; lsjk PDFBibTeX XMLCite \textit{V. V. Bytev} et al., Nucl. Phys., B 836, No. 3, 129--170 (2010; Zbl 1206.81089) Full Text: DOI arXiv References: [1] ’t Hooft, G.; Veltman, M., Nucl. Phys. B, 44, 189 (1972) [2] Anastasiou, C.; Melnikov, K., Nucl. Phys. B, 646, 220 (2002) [3] Regge, T., (DeWitt, C. M.; Wheeler, J. A., Battelle Rencontres: 1967 Lectures in Mathematics and Physics (1968), W.A. Benjamin: W.A. Benjamin New York), 433-458 [4] de Calan, C.; Malbouisson, A. P.C., Commun. Math. Phys., 90, 413 (1983) [5] Smirnov, V. A., Feynman Integral Calculus (2006), Springer: Springer Berlin [6] Del Duca, V.; Duhr, C.; Glover, E. W.N.; Smirnov, V. A., JHEP, 1001, 042 (2010) [7] Mano, K., Phys. Rev. D, 11, 452 (1975) [8] Gel’fand, I. M.; Graev, M. I.; Retakh, V. S., Russian Math. Sur., 47, 1 (1992) [9] (Erdelyi, A., Higher Transcendental Functions (1953), McGraw-Hill: McGraw-Hill New York) · Zbl 0051.30303 [10] Sato, M., Nagoya Math. J., 120, 1 (1990) [11] Kalmykov, M. Yu.; Bytev, V. V.; Kniehl, B. A.; Ward, B. F.L.; Yost, S. A., PoS(ACAT08), 125 (2008) [12] Takayama, N., Jpn. J. Appl. Math., 6, 147 (1989) [13] Gauss, C. F., (Gesammelte Werke, vol. 3 (1823), Teubner: Teubner Leipzig), 1866-1929 [14] Tarasov, O. V., Nucl. Instrum. Methods A, 534, 293 (2004) [15] Chetyrkin, K. G.; Tkachov, F. V., Nucl. Phys. B, 192, 159 (1981) [16] Kalmykov, M. Yu.; Ward, B. F.L.; Yost, S., JHEP, 0702, 040 (2007) [17] Yost, S. A.; Kalmykov, M. Yu.; Ward, B. F.L., in: Proceedings of the 34th International Conference in High Energy Physics (ICHEP08), Philadelphia, USA, 2008, eConf C080730 [18] Kalmykov, M. Yu.; Ward, B. F.L.; Yost, S., JHEP, 0711, 009 (2007) [19] Kalmykov, M. Yu.; Kniehl, B. A., Nucl. Phys. B, 809, 365 (2009) [20] Davydychev, A. I.; Kalmykov, M. Yu., Nucl. Phys. B, 699, 3 (2004) [21] Rainville, E. D., Bull. Amer. Math. Soc., 51, 714 (1945) [22] Takayama, N., J. Symbolic Comput., 20, 637 (1995) [23] Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I., Integrals and Series, More Special Functions, vol. 3 (1990), Gordon and Breach: Gordon and Breach New York · Zbl 0967.00503 [24] Gottschalk, J. E.; Maslen, E. N., J. Phys. A, 21, 1983 (1988) [25] Karlsson, P. W., J. Math. Phys., 12, 270 (1971) [26] Baikov, P. A., Phys. Lett. B, 634, 325 (2006) [27] Weinzierl, S., J. Math. Phys., 45, 2656 (2004) [28] Davydychev, A. I.; Kalmykov, M. Yu., Nucl. Phys. B (Proc. Suppl.), 89, 283 (2000) [29] Davydychev, A. I.; Kalmykov, M. Yu., Nucl. Phys. B, 605, 266 (2001) [30] Kalmykov, M. Yu.; Kniehl, B. A.; Ward, B. F.L.; Yost, S. A., in: V.A. Duk, V.A. Matveev, A.A. Rubakov (Eds.), Proceedings of the 15th International Seminar on High Energy Physics (QUARKS-2008), Sergiev Posad, Russia, 2008 [31] Kalmykov, M. Yu., Nucl. Phys. B (Proc. Suppl.), 135, 280 (2004) [32] Huber, T.; Maître, D., Comput. Phys. Commun., 178, 755 (2008) [33] Norlund, N. E., Acta Math., 94, 289 (1955) [34] Vollinga, J.; Weinzierl, S., Comput. Phys. Commun., 167, 177 (2005) [35] Kalmykov, M. Yu., JHEP, 0604, 056 (2006) [36] Bytev, V. V.; Kalmykov, M.; Kniehl, B. A.; Ward, B. F.L.; Yost, S. A., (Speer, T.; Carminati, F.; Peret-Gallix, D.; Brun, R., Proceedings of the International Linear Collider Workshop (LCWS08 and ILC08). Proceedings of the International Linear Collider Workshop (LCWS08 and ILC08), Chicago, USA, 2008 (2009), Proceedings of Science: Proceedings of Science Trieste) [37] Tarasov, O. V., Phys. Rev. D, 54, 6479 (1996) [38] Boos, E. E.; Davydychev, A. I., Theor. Math. Phys., 89, 1052 (1991) [39] Berends, F. A.; Davydychev, A. I.; Smirnov, V. A.; Tausk, J. B., Nucl. Phys. B, 439, 536 (1995) [40] Davydychev, A. I., Nucl. Instrum. Methods A, 559, 293 (2006) [41] Anastasiou, C.; Glover, E. W.N.; Oleari, C., Nucl. Phys. B, 572, 307 (2000) [42] Pivovarov, G. B.; Tkachov, F. V., Int. J. Mod. Phys. A, 8, 2241 (1993) [43] Avdeev, L. V., Comput. Phys. Commun., 98, 15 (1996) [44] Jegerlehner, F.; Kalmykov, M. Yu.; Veretin, O., Nucl. Phys. B, 641, 285 (2002) [45] Davydychev, A. I., Loop calculations in QCD with massive quarks, Presentation at International Conference on Relativistic Nuclear Dynamics, Vladivostok, Russia, 1991 · Zbl 0729.58054 [46] Broadhurst, D. J.; Fleischer, J.; Tarasov, O. V., Z. Phys. C, 60, 287 (1993) [47] Tarasov, O. V., Nucl. Phys. B, 502, 455 (1997) [48] Kalmykov, M. Yu. [49] Jegerlehner, F.; Kalmykov, M. Yu.; Veretin, O., Nucl. Phys. B (Proc. Suppl.), 116, 382 (2003) [50] Huber, T., Nucl. Phys. B (Proc. Suppl.), 183, 238 (2008) [51] Davydychev, A. I.; Tausk, J. B., Nucl. Phys. B, 397, 123 (1993) [52] Fleischer, J.; Kalmykov, M. Yu., Phys. Lett. B, 470, 168 (1999) [53] Czakon, M.; Awramik, M.; Freitas, A., Nucl. Phys. B (Proc. Suppl.), 157, 58 (2006) [54] Kniehl, B. A.; Tarasov, O. V., Nucl. Phys. B, 820, 178 (2009) [55] Bekavac, S.; Grozin, A. G.; Seidel, D.; Smirnov, V. A., Nucl. Phys. B, 819, 183 (2009) [56] Davydychev, A. I.; Grozin, A. G., Phys. Rev. D, 59, 054023 (1999) [57] Onishchenko, A.; Veretin, O., Phys. Atom. Nucl., 68, 1405 (2005) [58] Actis, S.; Passarino, G.; Sturm, C.; Uccirati, S., Nucl. Phys. B, 811, 182 (2009) [59] Frink, A.; Kniehl, B. A.; Kreimer, D.; Riesselmann, K., Phys. Rev. D, 54, 4548 (1996) [60] Kniehl, B. A.; Kotikov, A. V.; Veretin, O. L., Phys. Rev. A, 80, 052501 (2009) [61] Vladimirov, A. A., Theor. Math. Phys.. Theor. Math. Phys., Teor. Mat. Fiz., 43, 210 (1980) [62] Berends, F. A.; Buza, M.; Böhm, M.; Scharf, R., Z. Phys. C, 63, 227 (1994) [63] V.V. Bytev, M.Yu. Kalmykov, B.A. Kniehl, in preparation; V.V. Bytev, M.Yu. Kalmykov, B.A. Kniehl, in preparation [64] Chyzak, F., (Gröbner Bases, Symbolic Summation and Symbolic Integration. Gröbner Bases, Symbolic Summation and Symbolic Integration, London Math. Soc. Lecture Notes Ser., vol. 251 (1998), Cambridge University Press: Cambridge University Press Cambridge), 32-60 · Zbl 0898.68040 [65] P. Paule, Contiguous relations and creative telescoping, Technical Report, RISC, Austria, 2001; P. Paule, Contiguous relations and creative telescoping, Technical Report, RISC, Austria, 2001 [66] Laporta, S., Int. J. Mod. Phys. A, 15, 5087 (2000) [67] Borwein, J. M.; Broadhurst, D. J.; Kamnitzer, J., Exper. Math., 10, 25 (2001) [68] Kalmykov, M. Y.; Sheplyakov, A., Comput. Phys. Commun., 172, 45 (2005) [69] Lewin, L., Polylogarithms and Associated Functions (1981), North-Holland: North-Holland Amsterdam · Zbl 0465.33001 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.