Harmonic sums, polylogarithms, special numbers, and their generalizations. (English) Zbl 1308.81140
Schneider, Carsten (ed.) et al., Computer algebra in quantum field theory. Integration, summation and special functions. Wien: Springer (ISBN 978-3-7091-1615-9/hbk; 978-1-4614-8523-0/ebook). Texts and Monographs in Symbolic Computation, 1-32 (2013).
Summary: In these introductory lectures, we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and massive Feynman diagrams at higher loops. These quantities are elements of stuffle and shuffle algebras implying algebraic relations being widely independent of the special quantities considered. They are supplemented by structural relations. The generalizations are given in terms of generalized harmonic sums, (generalized) cyclotomic sums, and sums containing in addition binomial and inverse-binomial weights. To all these quantities iterated integrals and special numbers are associated. We also discuss the analytic continuation of nested sums of different kind to complex values of the external summation bound \(N\).
For the entire collection see [Zbl 1276.81004].
For the entire collection see [Zbl 1276.81004].
MSC:
| 81T18 | Feynman diagrams |
| 81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |
| 11G55 | Polylogarithms and relations with \(K\)-theory |
References:
| [1] | Lense, J., Reihenentwicklungen in der Mathematischen Physik (1953), Berlin: Walter de Gryter, Berlin · Zbl 0051.04802 · doi:10.1515/9783111499895 |
| [2] | Sommerfeld, A., Partielle Differentialgleichungen der Physik (1958), Vorlesungen über Theoreti- sche Physik, vol. VI. Akademische Verlagsgesellschaft Geest and Prtig, Leipzig: In, Vorlesungen über Theoreti- sche Physik, vol. VI. Akademische Verlagsgesellschaft Geest and Prtig, Leipzig |
| [3] | Forsyth, A.R.: Theory of Differential Equations, pp. 2-4. Cambridge University Press, Cambridge (1900-1902);Kamke, E.: Differentialgleichungen, Lösungsmethoden und Lösungen. Akademische Verlagsgesellschaft Geest and Portig, Leipzig (1967) |
| [4] | Kratzer, A.; Franz, W., Transzendente Funktionen (1963), Leipzig: Akademische Verlagsgesellschaft Geest and Portig, Leipzig |
| [5] | Andrews, G. E.; Askey, R.; Roy, R., Special Functions (2006), Cambridge: Cambridge University Press, Cambridge |
| [6] | Nakanishi, N.; Weinzierl, S., Feynman graph polynomials, Int. J. Mod. Phys. A, 18, 1-125 (1961) |
| [7] | Vermaseren, J. A.M.; Vogt, A.; Moch, S., The third-order QCD corrections to deep-inelastic scattering by photon exchange, Nucl. Phys. B, 724, 3-182 (2005) · Zbl 1178.81286 |
| [8] | Bierenbaum, I.; Blümlein, J.; Klein, S., Mellin moments of the O(α s 3) heavy flavor contributions to unpolarized deep-inelastic scattering at \[{Q}^2 \gg{m}^2\] and anomalous dimensions, Nucl. Phys. B, 820, 417-482 (2009) · Zbl 1194.81244 |
| [9] | Kontsevich, M.; Zagier, D.; Engquist, B.; Schmid, W., Periods. IMHS/M/01/22, Mathematics Unlimited - 2001 and Beyond, 771-808 (2011), Berlin: Springer, Berlin · Zbl 1039.11002 |
| [10] | Bogner, C.; Weinzierl, S., Periods and Feynman integrals, J. Math. Phys., 50, 042302 (2009) · Zbl 1214.81096 |
| [11] | Poincaré, H.; Lappo-Danilevsky, J. A.; Chen, K. T., Sur les groupes des équations linéaires, Acta Math., 156, 3, 201-312 (1884) · JFM 16.0252.01 |
| [12] | Jonquière, A., Über einige Transcendente, welche bei der wiederholten Integration rationaler Funktionen auftreten, Bihang till Kongl. Svenska Vetenskaps-Akademiens Handlingar, 15, 1-50 (1889) · JFM 21.0432.01 |
| [13] | Mellin, H., Über den Zusammenhang zwischen den linearen Differential- und Differenzengleichungen, Acta Soc. Fennicae, 21, 1-115 (1902) · JFM 32.0348.02 |
| [14] | Blümlein, J.: Structural relations of harmonic sums and Mellin transforms up to weight w = 5. Comput. Phys. Commun. 180, 2218-2249 (2009). [arXiv:0901.3106 [hep-ph]] · Zbl 1197.81036 |
| [15] | Blümlein, J.; Klein, S.; Schneider, C.; Stan, F., A symbolic summation approach to Feynman integral calculus, J. Symb. Comput., 47, 1267-1289 (2012) · Zbl 1242.81005 |
| [16] | Barnes, E.W.: A new development of the theory of the hypergeometric functions. Proc. Lond. Math. Soc. 6(2), 141 (1908); A transformation of generalized hypergeometric series. Quart. Journ. Math. 41, 136-140 (1910);Mellin, H.: Abriß einer einheitlichen Theorie der Gamma- und der hypergeometrischen Funktionen. Math. Ann. 68, 305-337 (1910) · JFM 41.0500.04 |
| [17] | Gluza, J.; Kajda, K.; Riemann, T., AMBRE: a mathematica package for the construction of Mellin-Barnes representations for Feynman integrals, Comput. Phys. Commun., 177, 879-893 (2007) · Zbl 1196.81131 |
| [18] | ’t Hooft, G., Veltman, M.J.G.: Regularization and renormalization of gauge fields. Nucl. Phys. B 44, 189-213 (1972);Bollini, C.G., Giambiagi, J.J.: Dimensional renormalization: the number of dimensions as a regularizing parameter. Nuovo Cim. B 12, 20-26 (1972);Ashmore, J.F.: A method of gauge invariant regularization. Lett. Nuovo Cim. 4, 289-290 (1972);Cicuta, G.M., Montaldi, E.: Analytic renormalization via continuous space dimension. Lett. Nuovo Cim. 4, 329-332 (1972) |
| [19] | Berends, F.A., van Neerven, W.L., Burgers, G.J.H.: Higher order radiative corrections at LEP energies. Nucl. Phys. B 297, 429-478 (1988). [Erratum-ibid. B 304, 921 (1988)];Blümlein, J., De Freitas, A., van Neerven, W.: Two-loop QED operator matrix elements with massive external fermion lines. Nucl. Phys. B 855, 508-569 (2012). [arXiv:1107.4638 [hep-ph]];Hamberg, R., van Neerven, W.L., Matsuura, T.: A complete calculation of the order α_s^2 correction to the Drell-Yan K-factor. Nucl. Phys. B 359, 343-405 (1991). [Erratum-ibid. B 644, 403-404 (2002)];Zijlstra, E.B., van Neerven, W.L.: Contribution of the second order gluonic Wilson coefficient to the deep inelastic structure function. Phys. Lett. B 273, 476-482 (1991); O(α_s^2) contributions to the deep inelastic Wilson coefficient. Phys. Lett. B 272, 127-133 (1991); O(α_s^2) QCD corrections to the deep inelastic proton structure functions F_2 and F_L. Nucl. Phys. B 383, 525-574 (1992); O(α_s^2) corrections to the polarized structure function g_1(x,Q^2). Nucl. Phys. B 417, 61-100 (1994). [Erratum-ibid. B 426, 245 (1994)], [Erratum-ibid. B 773, 105-106 (2007)];Kazakov, D.I., Kotikov, A.V.: Totalas correction to deep-inelastic scattering cross section ratio R = σ_L∕σ_T in QCD. Calculation of the longitudinal structure function. Nucl. Phys. B 307, 721-762 (1988). [Erratum-ibid. B 345, 299 (1990)];Kazakov, D.I., Kotikov, A.V., Parente, G., Sampayo, O.A., Sanchez Guillen, J.: Complete quartic (α_s^2) correction to the deep inelastic longitudinal structure function F_L in QCD. Phys. Rev. Lett. 65, 1535-1538 (1990). [Erratum-ibid. 65, 2921 (1990)];Sanchez Guillen, J., Miramontes, J., Miramontes, M., Parente, G., Sampayo, O.A.: Next-toleading order analysis of the deep inelastic R = σ_L∕σ_total. Nucl. Phys. B 353, 337-345 (1991) |
| [20] | Leibniz, G.W.: Mathematische Schriften. In: Gerhardt C.J. (ed.) vol. III, p. 351. Verlag H.W.Schmidt, Halle (1858);Euler, L.: Institutiones calculi integralis, vol. I, pp. 110-113. Impensis Academiae Imperialis Scientiarum, Petropoli (1768); Mémoires de l’Académie de Sint-P’etersbourg (1809-1810), vol. 3, pp. 26-42 (1811);Landen, J.: A new method of computing sums of certain series. Phil. Trans. R. Soc. Lond. 51, 553-565 (1760); Mathematical Memoirs, p. 112 (Printed for the Author, Nourse, J., London, 1780);Lewin, L.: Dilogarithms and Associated Functions. Macdonald, London (1958);Kirillov, A.N.: Dilogarithm identities. Prog. Theor. Phys. Suppl. 118, 61-142 (1995). [hep-th/9408113];Maximon, L.C.: The dilogarithm function for complex argument. Proc. R. Soc. A 459, 2807-2819 (2003);Zagier, D.: The remarkable dilogarithm. J. Math. Phys. Sci. 22, 131-145 (1988); In: Cartier, P., Julia, B., Moussa et al., P. (eds.) Frontiers in Number Theory, Physics, and Geometry II - On Conformal Field Theories, Discrete Groups and Renormalization, pp. 3-65. Springer, Berlin (2007) |
| [21] | Spence, W., An Essay of the Theory of the Various Orders of Logarithmic Transcendents (1809), London: John Murray, London |
| [22] | Kummer, E. E., Ueber die Transcendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen, J. für Math. (Crelle), 21, 74-90 (1840) · ERAM 021.0659cj · doi:10.1515/crll.1840.21.74 |
| [23] | Lewin, L.; Duke, D. W., Table of integrals and formulae for Feynman diagram calculations, Riv. Nuovo Cim., 7, 6, 1-39 (1984) |
| [24] | Nielsen, N.; Kölbig, K. S.; Mignoco, J. A.; Remiddi, E., On Nielsen’s generalized polylogarithms and their numerical calculation, BIT, 3, 125-211 (1909) |
| [25] | Vermaseren, J. A.M., Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A, 14, 2037-2076 (1999) · Zbl 0939.65032 |
| [26] | Blümlein, J.; Kurth, S., Harmonic sums and Mellin transforms up to two loop order, Phys. Rev. D, 60, 014018 (1999) |
| [27] | Remiddi, E.; Vermaseren, J. A.M., Harmonic polylogarithms, Int. J. Mod. Phys. A, 15, 725-754 (2000) · Zbl 0951.33003 |
| [28] | Moch, S.-O.; Uwer, P.; Weinzierl, S., Nested sums, expansion of transcendental functions and multiscale multiloop integrals, J. Math. Phys., 43, 3363-3386 (2002) · Zbl 1060.33007 |
| [29] | Ablinger, J., Blümlein, J., Schneider, C.: Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms. [arXiv:1302.0378 [math-ph]] · Zbl 1295.81071 |
| [30] | Ablinger, J.; Blümlein, J.; Schneider, C., Harmonic sums and polylogarithms generated by cyclotomic polynomials, J. Math. Phys., 52, 102301 (2011) · Zbl 1272.81127 |
| [31] | Laporta, S.: High precision \(\varepsilon \)-expansions of massive four loop vacuum bubbles Phys. Lett. B 549, 115-122 (2002). [hep-ph/0210336]; Analytical expressions of 3 and 4-loop sunrise Feynman integrals and 4-dimensional lattice integrals. Int. J. Mod. Phys. A 23, 5007-5020 (2008). [arXiv:0803.1007 [hep-ph]];Bailey, D.H., Borwein, J.M., Broadhurst, D., Glasser, M.L.: Elliptic integral evaluations of Bessel moments. [arXiv:0801.0891 [hep-th]];Müller-Stach, S., Weinzierl, S., Zayadeh, R.: Picard-Fuchs equations for Feynman integrals. [arXiv:1212.4389 [hep-ph]];Adams, L., Bogner, C., Weinzierl, S.: The two-loop sunrise graph with arbitrary masses. [arXiv:1302.7004 [hep-ph]] |
| [32] | Schneider, C.; Differ, J.; Carey, A.; Ellwood, D.; Paycha, S.; Rosenberg, S., Product representations in ΠΣ-fields, Equ. Appl., 6, 9, 365-386 (2004) |
| [33] | Bronstein, M., Symbolic Integration I: Transcendental Functions (1997), Berlin: Springer, Berlin · Zbl 0880.12005 · doi:10.1007/978-3-662-03386-9 |
| [34] | Raab, C.: Definite integration in differential fields. PhD thesis, Johannes Kepler University Linz (2012); and this volume |
| [35] | Ablinger, J.: A computer algebra toolbox for harmonic sums related to particle physics. Master’s thesis, Johannes Kepler University (2009). [arXiv:1011.1176 [math-ph]]; Computer algebra algorithms for special functions in particle physics. PhD thesis, Johannes Kepler University Linz (2012) |
| [36] | Feynman, R. P., Space - time approach to quantum electrodynamics, Phys. Rev., 76, 769-789 (1949) · Zbl 0038.13302 · doi:10.1103/PhysRev.76.769 |
| [37] | Napier, J.: Mirifici logarithmorum canonis descriptio, ejusque usus, in utraque trigonometria; ut etiam in omni logistica mathematica, amplissimi, facillimi, & expeditissimi explacatio. Andrew Hart, Edinburgh (1614) |
| [38] | Racah, G., Sopra l’rradiazione nell’urto di particelle veloci, Nuovo Com., 11, 461-476 (1934) · Zbl 0010.13802 · doi:10.1007/BF02959918 |
| [39] | Fleischer, J.; Kotikov, A. V.; Veretin, O. L., Analytic two loop results for selfenergy type and vertex type diagrams with one nonzero mass, Nucl. Phys. B, 547, 343-374 (1999) |
| [40] | Blümlein, J., Ravindran, V.: Mellin moments of the next-to-next-to leading order coefficient functions for the Drell-Yan process and hadronic Higgs-boson production. Nucl. Phys. B 716, 128-172 (2005). [hep-ph/0501178]; O(α_s^2) timelike Wilson coefficients for parton-fragmentation functions in Mellin space. Nucl. Phys. B 749, 1-24 (2006). [hep-ph/0604019];Blümlein, J., Klein, S.: Structural relations between harmonic sums up to w=6. PoS ACAT 084 (2007). [arXiv:0706.2426 [hep-ph]];Bierenbaum, I., Blümlein, J., Klein, S., Schneider, C.: Two-loop massive operator matrix elements for unpolarized heavy flavor production to \(O( \varepsilon )\). Nucl. Phys. B 803, 1-41 (2008). [arXiv:0803.0273 [hep-ph]];Czakon, M., Gluza, J., Riemann, T.: Master integrals for massive two-loop Bhabha scattering in QED. Phys. Rev. D 71, 073009 (2005). [hep-ph/0412164] |
| [41] | Moch, S.; Vermaseren, J. A.M.; Vogt, A., The three loop splitting functions in QCD: the non-singlet case, Nucl. Phys. B, 688, 101-134 (2004) · Zbl 1149.81371 |
| [42] | Gehrmann, T.; Remiddi, E., Numerical evaluation of harmonic polylogarithms, Comput. Phys. Commun., 141, 296-312 (2001) · Zbl 0991.65022 · doi:10.1016/S0010-4655(01)00411-8 |
| [43] | Vollinga, J.; Weinzierl, S., Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun., 167, 177-194 (2005) · Zbl 1196.65045 · doi:10.1016/j.cpc.2004.12.009 |
| [44] | Gonzalez-Arroyo, A.; Lopez, C.; Yndurain, F. J., Second order contributions to the structure functions in deep inelastic scattering. 1. Theoretical calculations. Nucl. Phys. B 153, 161-186 (1979);Floratos, E.G., Kounnas, C., Lacaze, R.: Higher order QCD effects in inclusive annihilation and deep inelastic scattering. Nucl. Phys, B, 192, 417-462 (1981) |
| [45] | Mertig, R.; Neerven, W. L., The calculation of the two loop spin splitting functions P ij (1)(x), Z. Phys. C, 70, 637-654 (1996) |
| [46] | Wilson, K. G.; Zimmermann, W.; Brandt, R. A.; Preparata, G.; Frishman, Y.; Blümlein, J., Non-lagrangian models of current algebra, Phys. Rev., 179, 1499-1512 (1969) |
| [47] | Hoffman, M. E., The Hopf algebra structure of multiple harmonic sums, Nucl. Phys. (Proc. Suppl.), 11, 49-68 (2000) · Zbl 0959.16021 |
| [48] | Berndt, B. C., Ramanujan’s Notebooks (1985), Springer, Berlin: Part I, Springer, Berlin · Zbl 0555.10001 · doi:10.1007/978-1-4612-1088-7 |
| [49] | Faà di Bruno, F.: Einleitung in die Theorie dier Binären Formen, deutsche Bearbeitung von Th. Walter. Teubner, Leipzig (1881) · JFM 13.0086.02 |
| [50] | Blümlein, J., Algebraic relations between harmonic sums and associated quantities, Comput. Phys. Commun., 159, 19-54 (2004) · Zbl 1097.11063 |
| [51] | Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; Lisonek, P., Special values of multiple polylogarithms, Trans. Am. Math. Soc., 353, 907-941 (2001) · Zbl 1002.11093 |
| [52] | Lyndon, R. C., On Burnsides problem, Trans. Am. Math. Soc., 77, 202-215 (1954) · Zbl 0058.01702 |
| [53] | Reutenauer, C., Free Lie algebras (1993), Oxford: University Press, Oxford · Zbl 0798.17001 |
| [54] | Radford, D. E., J. Algebra, 58, 432-454 (1979) · Zbl 0409.16011 · doi:10.1016/0021-8693(79)90171-6 |
| [55] | Witt, E., Die Unterringe der freien Lieschen Ringe, Math. Zeitschr., 177, 152-160 (1937) · JFM 63.0089.02 |
| [56] | Möbius, A.F.: Über eine besondere Art von Umkehrung der Reihen. J. Reine Angew. Math. 9, 105-123 (1832);Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Calendron Press, Oxford (1978) · ERAM 009.0333cj |
| [57] | Maitre, D., HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun., 174, 222-240 (2006) · Zbl 1196.68330 |
| [58] | Ablinger, J., Blümlein, J., Schneider, C.: DESY 13-064 |
| [59] | Blümlein, J., The theory of deeply inelastic scattering, Prog. Part. Nucl. Phys., 69, 28 (2013) |
| [60] | Blümlein, J.: Structural relations of harmonic sums and Mellin transforms at weight w = 6. In: Carey, A., Ellwood, D., Paycha, S., Rosenberg, S. (eds.) Motives, Quantum Field Theory, and Pseudodifferential Operators, vol. 12, pp. 167-186. Clay Mathematics Proceedings, American Mathematical Society (2010). [arXiv:0901.0837 [math-ph]]. · Zbl 1218.81103 |
| [61] | Euler, L., Meditationes circa singulare serium genus, Novi Commentarii academiae scientiarum Petropolitanae, 20, 140-186 (1776) |
| [62] | Zagier, D.: Values of zeta functions and their applications. In: First European Congress of Mathematics (Paris, 1992), vol. II. Prog. Math. 120, 497-512 (Birkhäuser, Basel-Boston) (1994) · Zbl 0822.11001 |
| [63] | Blümlein, J.; Broadhurst, D. J.; Vermaseren, J. A.M., The multiple zeta value data mine, Comput. Phys. Commun., 181, 582-625 (2010) · Zbl 1221.11183 |
| [64] | Hoffman, M.E.: Hoffman’s page. http://www.usna.edu/Users/math/ meh/biblio.html |
| [65] | .fr/ fischler/publi.html“ TargetType=”URL”/>, Valeurs zêta multiples. Une introduction, Journal de théorie des nombres de Bordeaux, 294, 2, 27-62 (2002) |
| [66] | Barbieri, R.; Mignaco, J. A.; Remiddi, E., Electron form-factors up to fourth order. 1. Nuovo Cim. A 11, 824-864 (1972);Levine, M.J., Remiddi, E., Roskies, R.: Analytic contributions to the G factor of the electron in sixth order. Phys. Rev, D, 20, 2068-2076 (1979) |
| [67] | Kuipers, J., Vermaseren, J.A.M.: About a conjectured basis for multiple zeta values. [arXiv:1105.1884 [math-ph]] |
| [68] | Goncharov, A.B.: Multiple polylogarithms and mixed Tate motives. [arxiv:math.AG/0103059];Terasoma, T.: Mixed Tate motives and multiple zeta values. Invent. Math. 149(2), 339-369 (2002). arxiv:math.AG/010423;Deligne, P., Goncharov, A.B.: Groupes fondamentaux motiviques de Tate mixtes. Ann. Sci. Ecole Norm. Sup. Série IV 38(1), 1-56 (2005) |
| [69] | Broadhurst, D.J.: On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory. [arXiv:hep-th/9604128] |
| [70] | Broadhurst, D. J.; Kreimer, D., Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B, 393, 403-412 (1997) · Zbl 0946.81028 · doi:10.1016/S0370-2693(96)01623-1 |
| [71] | http://en.wikipedia.org/wiki/Padovan_sequence;http://www.emis.de/journals/NNJ/conferences/N2002-Padovan.html |
| [72] | Perrin, R.; Williams, A.; Shanks, D., Strong primality tests that are not sufficient, Math. Comput., 39, 159, 76-77 (1899) |
| [73] | de filiis Bonaccij, L.P.: Liber abaci, Cap. 12.7. Pisa (1202);Sigler, L.E.: Fibonacci’s Liber Abaci. Springer, Berlin (2002) |
| [74] | Hoffman, M. E., The algebra of multiple harmonic series, J. Algebra, 194, 477-495 (1997) · Zbl 0881.11067 · doi:10.1006/jabr.1997.7127 |
| [75] | Brown, F., Mixed tate motives over Z, Ann. Math., 175, 1, 949-976 (2012) · Zbl 1278.19008 · doi:10.4007/annals.2012.175.2.10 |
| [76] | The formula goes back to de Moivre, Bernoulli, Euler, and later Binet, see: Beutelspacher, A., Petri, B.: Der Goldene Schnitt. Spektrum, Heidelberg (1988) |
| [77] | Lucas, E., Théorie des fonctions numériques simplement périodiques, Am. J. Math., 1, 197-240 (1878) · doi:10.2307/2369311 |
| [78] | Euler, L.: Meditationes circa singulare serium genus. Novi Comm. Acad. Sci. Petropol. 20, 140-186 (1775). (reprinted in Opera Omnia Ser I, vol. 15, pp. 217-267. B.G. Teubner, Berlin (1927)) |
| [79] | Hoffman, M. E.; Moen, C.; Granville, A.; Motohashi, Y., Sums of triple harmonic series, J. Number Theory, 60, 329-331 (1996) · Zbl 0863.11051 |
| [80] | Hoffman, M. E., Multiple harmonic series, Pac. J. Math., 152, 275-290 (1992) · Zbl 0763.11037 · doi:10.2140/pjm.1992.152.275 |
| [81] | Hoffman, M.E.: Algebraic aspects of multiple zeta values. In: Aoki et al., T. (eds.) Zeta Functions, Topology and Quantum Physics. Developments in Mathematics, vol. 14, pp. 51-74. Springer, New York (2005). [arXiv:math/0309452 [math.QA]] · Zbl 1170.11324 |
| [82] | Okuda, J., Ueno, K.: The sum formula of multiple zeta values and connection problem of the formal Knizhnik-Zamolodchikov equation. In: Aoki et al., T. (eds.) Zeta Functions, Topology and Quantum Physics. Developments in Mathematics, vol. 14, pp. 145-170. Springer, New York (2005). [arXiv:math/0310259 [math.NT]] · Zbl 1170.11327 |
| [83] | Hoffmann, M. E.; Ohno, Y., Relations of multiple zeta values and their algebraic expressions, J. Algebra, 262, 332-347 (2003) · Zbl 1139.11322 · doi:10.1016/S0021-8693(03)00016-4 |
| [84] | Zudilin, V.V.: Algebraic relations for multiple zeta values. Uspekhi Mat. Nauk 58(1), 3-22 |
| [85] | Ihara, K., Kaneko, M.: A note on relations of multiple zeta values (preprint) · Zbl 1186.11053 |
| [86] | Le, T. Q.T.; Murakami, J., Kontsevich’s integral for the Homfly polynomial and relations between values of multiple zeta functions, Topol. Appl., 62, 193-206 (1995) · Zbl 0839.57007 · doi:10.1016/0166-8641(94)00054-7 |
| [87] | Ohno, Y., A generalization of the duality and sum formulas on the multiple zeta values, J. Number Theory, 74, 189-209 (1999) · Zbl 0920.11063 · doi:10.1006/jnth.1998.2314 |
| [88] | Ohno, Y., Zagier, D.: Indag. Math. (N.S.) 12, 483-487 (2001) · Zbl 1031.11053 |
| [89] | Ohno, Y.; Wakabayashi, N., Cyclic sum of multiple zeta values, Acta Arithmetica, 123, 289-295 (2006) · Zbl 1156.11038 · doi:10.4064/aa123-3-5 |
| [90] | Borwein, J.M., Bradley, D.M., Broadhurst, D.J.: Evaluations of K fold Euler/Zagier sums: a compendium of results for arbitrary k. [hep-th/9611004] · Zbl 0884.40004 |
| [91] | Ihara, K.; Kaneko, M.; Zagier, D., Derivation and double shuffle relations for multiple zeta values, Compositio Math., 142, 307-338 (2006) · Zbl 1186.11053 |
| [92] | Écalle, J.: Théorie des moules, vol. 3, prépublications mathématiques d‘Orsay (1981, 1982, 1985); La libre génération des multicêtas et leur d’ecomposition canonico-explicite en irréductibles, automne (1999); Ari/gari et la décomposition des multizêtas en irréductibles. Prépublication, avril (2000) |
| [93] | Berndt, B.C.: Ramanujan’s Notebooks, Part IV, pp. 323-326. Springer, New York (1994) · Zbl 0785.11001 |
| [94] | Ablinger, J., Blümlein, J., Raab, C., Schneider, C., Wißbrock, F.: DESY 13-063 |
| [95] | Weinzierl, S.: Feynman Graphs. [arXiv:1301.6918 [hep-ph]] |
| [96] | Hopf, H., Über die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeinerungen. Ann. Math. 42, 22-52 (1941);Milner, J., Moore, J.: On the structure of Hopf algebras. Ann. Math. 81, 211-264 (1965);Sweedler (1969), Hopf Algebras. Benjamin, New York: M.E, Hopf Algebras. Benjamin, New York |
| [97] | Goncharov, A. B., Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett., 5, 497-516 (1998) · Zbl 0961.11040 |
| [98] | Gehrmann, T.; Remiddi, E., Two loop master integrals for γ∗ → 3 jets: the planar topologies, Nucl. Phys. B, 601, 248-286 (2001) |
| [99] | Weinzierl, S., Symbolic expansion of transcendental functions, Comput. Phys. Commun., 145, 357-370 (2002) · Zbl 1001.65025 |
| [100] | Moch, S.-O.; Uwer, P., XSummer: transcendental functions and symbolic summation in FORM, Comput. Phys. Commun., 174, 759-770 (2006) · Zbl 1196.68332 |
| [101] | Appell, P.; Gauthier-Villars, Paris; Appell, P.; Kampé, J.; Fériet, Paris; Gauthier-Villars, W. N.; Bailey, L. J.; Slater, C.; Glover, E. W.N.; Oleari, C., Application of the negative dimension approach to massless scalar box integrals, Nucl. Phys. B, 565, 445-467 (1925) · Zbl 0956.81053 |
| [102] | Ablinger, J.; Blümlein, J.; Hasselhuhn, A.; Klein, S.; Schneider, C.; Wißbrock, F., Massive 3-loop ladder diagrams for quarkonic local operator matrix elements, Nucl. Phys. B, 864, 52-84 (2012) · Zbl 1262.81184 |
| [103] | Ablinger, J., et al.: New results on the 3-loop heavy Flavor Wilson coefficients in deep-inelastic scattering. [arXiv:1212.5950 [hep-ph]]; Three-loop contributions to the gluonic massive operator matrix elements at general values of N. PoS LL 2012, 033 (2012). [arXiv:1212.6823 [hep-ph]] |
| [104] | Lang, S., Algebra (2002), New York: Springer, New York · Zbl 0984.00001 · doi:10.1007/978-1-4613-0041-0 |
| [105] | Euler, L.: Theoremata arithmetica nova methodo demonstrata. Novi Commentarii academiae scientiarum imperialis Petropolitanae, vol. 8, pp. 74-104 (1760/1, 1763); Opera Omnia, Ser. I, vol. 2, pp. 531-555. Takase, M.: Euler’s Theory of Numbers In: Baker, R. (ed.) Euler Reconsidered, pp. 377-421. Kedrick Press, Heber City (2007). leonhardeuler.web.fc2.com/eulernumber_en.pdf |
| [106] | Catalan, E.: Recherches sur la constant G, et sur les integrales euleriennes. Mémoires de l’Academie imperiale des sciences de Saint-Pétersbourg, Ser. 7(31), 1-51 (1883);Adamchik, V. http://www-2.cs.cmu.edu/ adamchik/articles/catalan/catalan.htm · JFM 15.0229.02 |
| [107] | Broadhurst, D. J., Massive three - loop Feynman diagrams reducible to SC^∗ primitives of algebras of the sixth root of unity, Eur. Phys. J. C, 8, 311-333 (1999) |
| [108] | Racinet, G.: Torseurs associes a certaines relations algebriques entre polyzetas aux racines de l’unite. Comptes rendus de l’Académie des sciences. Série 1, Mathématique 333(1), 5-10 (2001). [arXiv:math.QA/0012024] · Zbl 1033.11033 |
| [109] | -th/0303162];Weinzierl, S., Massive Feynman diagrams and inverse binomial sums, Nucl. Phys. B, 699, 3-64 (2004) · Zbl 1123.81388 |
| [110] | Ablinger, J., Blümlein, J., Raab, C., Schneider, C.: (in preparation) |
| [111] | Aglietti, U., Bonciani, R.: Master integrals with 2 and 3 massive propagators for the 2 loop electroweak form-factor - planar case. Nucl. Phys. B 698, 277-318 (2004). [hep-ph/0401193];Bonciani, R., Degrassi, G., Vicini, A.: On the generalized harmonic polylogarithms of one complex variable. Comput. Phys. Commun. 182, 1253-1264 (2011). [arXiv:1007.1891 [hep-ph]] · Zbl 1262.65035 |
| [112] | Symanzik, K., Small distance behavior in field theory and power counting. Commun. Math. Phys. 18, 227-246 (1970);Callan, C.G., Jr.: Broken scale invariance in scalar field theory. Phys. Rev, D, 2, 1541-1547 (1970) · Zbl 0195.55902 |
| [113] | Blümlein, J.; Hasselhuhn, A.; Kovacikova, P.; Moch, S., O(α s ) heavy flavor corrections to charged current deep-inelastic scattering in Mellin space, Phys. Lett. B, 700, 294-304 (2011) |
| [114] | Blümlein, J.; Vogt, A., The evolution of unpolarized singlet structure functions at small x, Phys. Rev. D, 58, 014020 (1998) |
| [115] | Nielsen, N.: Handbuch der Theorie der Gammafunktion. Teubner, Leipzig (1906); Reprinted by Chelsea Publishing Company, Bronx, New York (1965) · JFM 37.0450.01 |
| [116] | Landau, E., Über die Grundlagen der Theorie der Fakultätenreihen, S.-Ber. math.-naturw. Kl. Bayerische Akad. Wiss. München, 36, 151-218 (1906) · JFM 37.0278.04 |
| [117] | Carlson, F.D.: Sur une classe de séries de Taylor. PhD thesis, Uppsala University (1914);see also http://en.wikipedia.org/wiki/Carlson/ · JFM 45.0645.01 |
| [118] | Blümlein, J.: Analytic continuation of Mellin transforms up to two loop order. Comput. Phys. Commun. 133, 76-104 (2000). [hep-ph/0003100];Blümlein, J., Moch, S.-O.: Analytic continuation of the harmonic sums for the 3-loop anomalous dimensions. Phys. Lett. B 614, 53-61 (2005). [hep-ph/0503188] |
| [119] | Kotikov, A.V., Velizhanin, V.N.: Analytic continuation of the Mellin moments of deep inelastic structure functions. [hep-ph/0501274] |
| [120] | Blümlein, J., Riemann, T., Schneider, C.: DESY Annual Report (2013) |
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.
