Harmonic Sums, Polylogarithms,Special Numbers, and Their Generalizations

Zurück zum Zitat Lense, J.: Reihenentwicklungen in der Mathematischen Physik. Walter de Gryter, Berlin (1953)CrossRefMATH Lense, J.: Reihenentwicklungen in der Mathematischen Physik. Walter de Gryter, Berlin (1953)CrossRefMATH

Zurück zum Zitat Sommerfeld, A.: Partielle Differentialgleichungen der Physik. In: Vorlesungen über Theoreti- sche Physik, vol. VI. Akademische Verlagsgesellschaft Geest and Prtig, Leipzig (1958) Sommerfeld, A.: Partielle Differentialgleichungen der Physik. In: Vorlesungen über Theoreti- sche Physik, vol. VI. Akademische Verlagsgesellschaft Geest and Prtig, Leipzig (1958)

Zurück zum Zitat 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) 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)

Zurück zum Zitat Kratzer, A., Franz, W.: Transzendente Funktionen. Akademische Verlagsgesellschaft Geest and Portig, Leipzig (1963) Kratzer, A., Franz, W.: Transzendente Funktionen. Akademische Verlagsgesellschaft Geest and Portig, Leipzig (1963)

Zurück zum Zitat Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (2006) Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (2006)

Zurück zum Zitat Nakanishi, N.: Parametric integral formulas and analytic properties in perturbation theory. Suppl. Progr. Theor. Phys. 18, 1–125 (1961); Graph Theory and Feynman Integrals. Gordon and Breach, New York (1970);Bogner, C., Weinzierl, S.: Feynman graph polynomials. Int. J. Mod. Phys. A 25, 2585–2618 (2010). [arXiv:1002.3458 [hep-ph]];Weinzierl, S.: (this volume) Nakanishi, N.: Parametric integral formulas and analytic properties in perturbation theory. Suppl. Progr. Theor. Phys. 18, 1–125 (1961); Graph Theory and Feynman Integrals. Gordon and Breach, New York (1970);Bogner, C., Weinzierl, S.: Feynman graph polynomials. Int. J. Mod. Phys. A 25, 2585–2618 (2010). [arXiv:1002.3458 [hep-ph]];Weinzierl, S.: (this volume)

Zurück zum Zitat 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). [arXiv:hep-ph/0504242] and refences therein 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). [arXiv:hep-ph/0504242] and refences therein

Zurück zum Zitat 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). [arXiv: 0904.3563 [hep-ph]] 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). [arXiv: 0904.3563 [hep-ph]]

Zurück zum Zitat Kontsevich, M., Zagier, D.: Periods. IMHS/M/01/22. In: Engquist, B., Schmid, W. (eds.) Mathematics Unlimited – 2001 and Beyond, pp. 771–808. Springer, Berlin (2011) Kontsevich, M., Zagier, D.: Periods. IMHS/M/01/22. In: Engquist, B., Schmid, W. (eds.) Mathematics Unlimited – 2001 and Beyond, pp. 771–808. Springer, Berlin (2011)

Zurück zum Zitat Bogner, C., Weinzierl, S.: Periods and Feynman integrals. J. Math. Phys. 50, 042302 (2009). [arXiv:0711.4863 [hep-th]] Bogner, C., Weinzierl, S.: Periods and Feynman integrals. J. Math. Phys. 50, 042302 (2009). [arXiv:0711.4863 [hep-th]]

Zurück zum Zitat Poincaré, H.: Sur les groupes des équations linéaires. Acta Math. 4, 201–312 (1884);Lappo-Danilevsky, J.A.: Mémoirs sur la Théorie des Systèmes Différentielles Linéaires. Chelsea Publishing Company, New York (1953);Chen, K.T.: Algebras of iterated path integrals and fundamental groups. Trans. A.M.S. 156(3), 359–379 (1971) Poincaré, H.: Sur les groupes des équations linéaires. Acta Math. 4, 201–312 (1884);Lappo-Danilevsky, J.A.: Mémoirs sur la Théorie des Systèmes Différentielles Linéaires. Chelsea Publishing Company, New York (1953);Chen, K.T.: Algebras of iterated path integrals and fundamental groups. Trans. A.M.S. 156(3), 359–379 (1971)

Zurück zum Zitat Jonquière, A.: Über einige Transcendente, welche bei der wiederholten Integration rationaler Funktionen auftreten. Bihang till Kongl. Svenska Vetenskaps-Akademiens Handlingar 15, 1–50 (1889) Jonquière, A.: Über einige Transcendente, welche bei der wiederholten Integration rationaler Funktionen auftreten. Bihang till Kongl. Svenska Vetenskaps-Akademiens Handlingar 15, 1–50 (1889)

Zurück zum Zitat Mellin, H.: Über die fundamentale Wichtigkeit des Satzes von Cauchy für die Theorien der Gamma- und hypergeometrischen Funktionen. Acta Soc. Fennicae 21, 1–115 (1886); Über den Zusammenhang zwischen den linearen Differential- und Differenzengleichungen. Acta Math. 25, 139–164 (1902) Mellin, H.: Über die fundamentale Wichtigkeit des Satzes von Cauchy für die Theorien der Gamma- und hypergeometrischen Funktionen. Acta Soc. Fennicae 21, 1–115 (1886); Über den Zusammenhang zwischen den linearen Differential- und Differenzengleichungen. Acta Math. 25, 139–164 (1902)

Zurück zum Zitat 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]] 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]]

Zurück zum Zitat Blümlein, J., Klein, S., Schneider, C., Stan, F.: A symbolic summation approach to Feynman integral calculus. J. Symb. Comput. 47, 1267–1289 (2012). [arXiv:1011.2656 [cs.SC]] Blümlein, J., Klein, S., Schneider, C., Stan, F.: A symbolic summation approach to Feynman integral calculus. J. Symb. Comput. 47, 1267–1289 (2012). [arXiv:1011.2656 [cs.SC]]

Zurück zum Zitat 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) 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)

Zurück zum Zitat 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). [arXiv:0704.2423 [hep-ph]] 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). [arXiv:0704.2423 [hep-ph]]

Zurück zum Zitat ’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) ’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)

Zurück zum Zitat 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) 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)

Zurück zum Zitat 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) 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)

Zurück zum Zitat Spence, W.: An Essay of the Theory of the Various Orders of Logarithmic Transcendents. John Murray, London (1809) Spence, W.: An Essay of the Theory of the Various Orders of Logarithmic Transcendents. John Murray, London (1809)

Zurück zum Zitat Kummer, E.E.: Ueber die Transcendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen. J. für Math. (Crelle) 21, 74–90 (1840)CrossRefMATH Kummer, E.E.: Ueber die Transcendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen. J. für Math. (Crelle) 21, 74–90 (1840)CrossRefMATH

Zurück zum Zitat Lewin, L.: Polylogarithms and Associated Functions. North Holland, New York (1981);Devoto, A., Duke, D.W.: Table of integrals and formulae for Feynman diagram calculations. Riv. Nuovo Cim. 7(6), 1–39 (1984) Lewin, L.: Polylogarithms and Associated Functions. North Holland, New York (1981);Devoto, A., Duke, D.W.: Table of integrals and formulae for Feynman diagram calculations. Riv. Nuovo Cim. 7(6), 1–39 (1984)

Zurück zum Zitat Nielsen, N.: Der Eulersche Dilogarithmus und seine Verallgemeinerungen. Nova Acta Leopold. XC Nr. 3, 125–211 (1909);Kölbig, K.S., Mignoco, J.A., Remiddi, E.: On Nielsen’s generalized polylogarithms and their numerical calculation. BIT 10, 38–74 (1970);Kölbig, K.S.: Nielsen’s generalized polylogarithms. SIAM J. Math. Anal. 17, 1232–1258 (1986) Nielsen, N.: Der Eulersche Dilogarithmus und seine Verallgemeinerungen. Nova Acta Leopold. XC Nr. 3, 125–211 (1909);Kölbig, K.S., Mignoco, J.A., Remiddi, E.: On Nielsen’s generalized polylogarithms and their numerical calculation. BIT 10, 38–74 (1970);Kölbig, K.S.: Nielsen’s generalized polylogarithms. SIAM J. Math. Anal. 17, 1232–1258 (1986)

Zurück zum Zitat Vermaseren, J.A.M.: Harmonic sums, Mellin transforms and integrals. Int. J. Mod. Phys. A 14, 2037–2076 (1999). [hep-ph/9806280] Vermaseren, J.A.M.: Harmonic sums, Mellin transforms and integrals. Int. J. Mod. Phys. A 14, 2037–2076 (1999). [hep-ph/9806280]

Zurück zum Zitat Blümlein, J., Kurth, S.: Harmonic sums and Mellin transforms up to two loop order. Phys. Rev. D 60, 014018 (1999). [hep-ph/9810241] Blümlein, J., Kurth, S.: Harmonic sums and Mellin transforms up to two loop order. Phys. Rev. D 60, 014018 (1999). [hep-ph/9810241]

Zurück zum Zitat Remiddi, E., Vermaseren, J.A.M.: Harmonic polylogarithms. Int. J. Mod. Phys. A 15, 725–754 (2000). [hep-ph/9905237] Remiddi, E., Vermaseren, J.A.M.: Harmonic polylogarithms. Int. J. Mod. Phys. A 15, 725–754 (2000). [hep-ph/9905237]

Zurück zum Zitat Moch, S.-O., Uwer, P., Weinzierl, S.: Nested sums, expansion of transcendental functions and multiscale multiloop integrals. J. Math. Phys. 43, 3363–3386 (2002). [hep-ph/0110083] Moch, S.-O., Uwer, P., Weinzierl, S.: Nested sums, expansion of transcendental functions and multiscale multiloop integrals. J. Math. Phys. 43, 3363–3386 (2002). [hep-ph/0110083]

Zurück zum Zitat Ablinger, J., Blümlein, J., Schneider, C.: Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms. [arXiv:1302.0378 [math-ph]] Ablinger, J., Blümlein, J., Schneider, C.: Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms. [arXiv:1302.0378 [math-ph]]

Zurück zum Zitat Ablinger, J., Blümlein, J., Schneider, C.: Harmonic sums and polylogarithms generated by cyclotomic polynomials. J. Math. Phys. 52, 102301 (2011). [arXiv:1105.6063 [math-ph]] Ablinger, J., Blümlein, J., Schneider, C.: Harmonic sums and polylogarithms generated by cyclotomic polynomials. J. Math. Phys. 52, 102301 (2011). [arXiv:1105.6063 [math-ph]]

Zurück zum Zitat 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]] 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]]

Zurück zum Zitat Schneider, C.: The summation package sigma: underlying principles and a rhombus tiling application. Discret. Math. Theor. Comput. Sci. 6, 365–386 (2004); Solving parameterized linear difference equations in terms of indefinite nested sums and products. Differ, J.: Equ. Appl. 11(9), 799–821 (2005); A new sigma approach to multi-summation. Adv. Appl. Math. 34(4), 740–767 (2005); Product representations in ΠΣ-fields. Ann. Comb. 9(1), 75–99 (2005); Symbolic summation assists combinatorics. Sem. Lothar. Combin. 56, 1–36 (2007); A refined difference field theory for symbolic summation. J. Symb. Comp. 43(9), 611–644 (2008). arXiv:0808.2543 [cs.SC]; Parameterized telescoping proves algebraic independence of sums. Ann. Comb. 14(4), 533–552 (2010). [arXiv:0808.2596 [cs.SC]]; Structural theorems for symbolic summation. Appl. Algebra Eng. Comm. Comput. 21(1), 1–32 (2010); A symbolic summation approach to find optimal nested sum representations. In: Carey, A., Ellwood, D., Paycha, S., Rosenberg, S. (eds.) Motives, Quantum Field Theory, and Pseudodifferential Operators, vol. 12, pp. 285–308. Clay Mathematics Proceedings. American Mathematical Society (2010). [arXiv:0904.2323 [cs.SC]]; and this volume Schneider, C.: The summation package sigma: underlying principles and a rhombus tiling application. Discret. Math. Theor. Comput. Sci. 6, 365–386 (2004); Solving parameterized linear difference equations in terms of indefinite nested sums and products. Differ, J.: Equ. Appl. 11(9), 799–821 (2005); A new sigma approach to multi-summation. Adv. Appl. Math. 34(4), 740–767 (2005); Product representations in ΠΣ-fields. Ann. Comb. 9(1), 75–99 (2005); Symbolic summation assists combinatorics. Sem. Lothar. Combin. 56, 1–36 (2007); A refined difference field theory for symbolic summation. J. Symb. Comp. 43(9), 611–644 (2008). arXiv:0808.2543 [cs.SC]; Parameterized telescoping proves algebraic independence of sums. Ann. Comb. 14(4), 533–552 (2010). [arXiv:0808.2596 [cs.SC]]; Structural theorems for symbolic summation. Appl. Algebra Eng. Comm. Comput. 21(1), 1–32 (2010); A symbolic summation approach to find optimal nested sum representations. In: Carey, A., Ellwood, D., Paycha, S., Rosenberg, S. (eds.) Motives, Quantum Field Theory, and Pseudodifferential Operators, vol. 12, pp. 285–308. Clay Mathematics Proceedings. American Mathematical Society (2010). [arXiv:0904.2323 [cs.SC]]; and this volume

Zurück zum Zitat Bronstein, M.: Symbolic Integration I: Transcendental Functions. Springer, Berlin (1997)CrossRefMATH Bronstein, M.: Symbolic Integration I: Transcendental Functions. Springer, Berlin (1997)CrossRefMATH

Zurück zum Zitat Raab, C.: Definite integration in differential fields. PhD thesis, Johannes Kepler University Linz (2012); and this volume Raab, C.: Definite integration in differential fields. PhD thesis, Johannes Kepler University Linz (2012); and this volume

Zurück zum Zitat 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) 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)

Zurück zum Zitat Feynman, R.P.: Space - time approach to quantum electrodynamics. Phys. Rev. 76, 769–789 (1949)MathSciNetADSCrossRefMATH Feynman, R.P.: Space - time approach to quantum electrodynamics. Phys. Rev. 76, 769–789 (1949)MathSciNetADSCrossRefMATH

Zurück zum Zitat 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) 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)

Zurück zum Zitat Racah, G.: Sopra l’rradiazione nell’urto di particelle veloci. Nuovo Com. 11, 461–476 (1934)CrossRef Racah, G.: Sopra l’rradiazione nell’urto di particelle veloci. Nuovo Com. 11, 461–476 (1934)CrossRef

Zurück zum Zitat 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). [hep-ph/9808242] 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). [hep-ph/9808242]

Zurück zum Zitat 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] 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]

Zurück zum Zitat 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). [hep-ph/0403192]; The three-loop splitting functions in QCD: the Singlet case. Nucl. Phys. B 691, 129–181 (2004). [hep-ph/0404111] 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). [hep-ph/0403192]; The three-loop splitting functions in QCD: the Singlet case. Nucl. Phys. B 691, 129–181 (2004). [hep-ph/0404111]

Zurück zum Zitat Gehrmann, T., Remiddi, E.: Numerical evaluation of harmonic polylogarithms. Comput. Phys. Commun. 141, 296–312 (2001). [arXiv:hep-ph/0107173]MathSciNetADSCrossRefMATH Gehrmann, T., Remiddi, E.: Numerical evaluation of harmonic polylogarithms. Comput. Phys. Commun. 141, 296–312 (2001). [arXiv:hep-ph/0107173]MathSciNetADSCrossRefMATH

Zurück zum Zitat Vollinga, J., Weinzierl, S.: Numerical evaluation of multiple polylogarithms. Comput. Phys. Commun. 167, 177–194 (2005). [arXiv:hep-ph/0410259]MathSciNetADSCrossRefMATH Vollinga, J., Weinzierl, S.: Numerical evaluation of multiple polylogarithms. Comput. Phys. Commun. 167, 177–194 (2005). [arXiv:hep-ph/0410259]MathSciNetADSCrossRefMATH

Zurück zum Zitat 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) 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)

Zurück zum Zitat Mertig, R., van Neerven, W.L.: The calculation of the two loop spin splitting functions P ij (1)(x). Z. Phys. C 70, 637–654 (1996). [hep-ph/9506451] Mertig, R., van Neerven, W.L.: The calculation of the two loop spin splitting functions P _ij ⁽¹⁾(x). Z. Phys. C 70, 637–654 (1996). [hep-ph/9506451]

Zurück zum Zitat Wilson, K.G.: Non-lagrangian models of current algebra. Phys. Rev. 179, 1499–1512 (1969); Zimmermann, W.: Lecture on Elementary Particle Physics and Quantum Field Theory, Brandeis Summer Institute, vol. 1, p. 395. MIT Press, Cambridge (1970); Brandt, R.A., Preparata, G.: The light cone and photon-hadron interactions. Fortsch. Phys. 20, 571–594 (1972); Frishman, Y.: Operator products at almost light like distances. Ann. Phys. 66, 373–389 (1971); Blümlein, J., Kochelev, N.: On the twist-2 and twist-three contributions to the spin dependent electroweak structure functions. Nucl. Phys. B 498, 285–309 (1997). [hep-ph/9612318] Wilson, K.G.: Non-lagrangian models of current algebra. Phys. Rev. 179, 1499–1512 (1969); Zimmermann, W.: Lecture on Elementary Particle Physics and Quantum Field Theory, Brandeis Summer Institute, vol. 1, p. 395. MIT Press, Cambridge (1970); Brandt, R.A., Preparata, G.: The light cone and photon-hadron interactions. Fortsch. Phys. 20, 571–594 (1972); Frishman, Y.: Operator products at almost light like distances. Ann. Phys. 66, 373–389 (1971); Blümlein, J., Kochelev, N.: On the twist-2 and twist-three contributions to the spin dependent electroweak structure functions. Nucl. Phys. B 498, 285–309 (1997). [hep-ph/9612318]

Zurück zum Zitat Hoffman, M.E.: Quasi-shuffle products. J. Algebraic Combin. 11, 49–68 (2000). [arXiv:math/9907173 [math.QA]]; The Hopf algebra structure of multiple harmonic sums. Nucl. Phys. (Proc. Suppl.) 135, 215 (2004). [arXiv:math/0406589] Hoffman, M.E.: Quasi-shuffle products. J. Algebraic Combin. 11, 49–68 (2000). [arXiv:math/9907173 [math.QA]]; The Hopf algebra structure of multiple harmonic sums. Nucl. Phys. (Proc. Suppl.) 135, 215 (2004). [arXiv:math/0406589]

Zurück zum Zitat Berndt, B.C.: Ramanujan’s Notebooks, Part I. Springer, Berlin (1985)CrossRefMATH Berndt, B.C.: Ramanujan’s Notebooks, Part I. Springer, Berlin (1985)CrossRefMATH

Zurück zum Zitat Faà di Bruno, F.: Einleitung in die Theorie dier Binären Formen, deutsche Bearbeitung von Th. Walter. Teubner, Leipzig (1881) Faà di Bruno, F.: Einleitung in die Theorie dier Binären Formen, deutsche Bearbeitung von Th. Walter. Teubner, Leipzig (1881)

Zurück zum Zitat Blümlein, J.: Algebraic relations between harmonic sums and associated quantities. Comput. Phys. Commun. 159, 19–54 (2004). [hep-ph/0311046] Blümlein, J.: Algebraic relations between harmonic sums and associated quantities. Comput. Phys. Commun. 159, 19–54 (2004). [hep-ph/0311046]

Zurück zum Zitat Borwein, J.M., Bradley, D.M., Broadhurst, D.J., Lisonek, P.: Special values of multiple polylogarithms. Trans. Am. Math. Soc. 353, 907–941 (2001). [math/9910045 [math-ca]] Borwein, J.M., Bradley, D.M., Broadhurst, D.J., Lisonek, P.: Special values of multiple polylogarithms. Trans. Am. Math. Soc. 353, 907–941 (2001). [math/9910045 [math-ca]]

Zurück zum Zitat Lyndon, R.C.: On Burnsides problem. Trans. Am. Math. Soc. 77, 202–215 (1954); On Burnsides problem II. Trans. Amer. Math. Soc. 78, 329–332 (1955) Lyndon, R.C.: On Burnsides problem. Trans. Am. Math. Soc. 77, 202–215 (1954); On Burnsides problem II. Trans. Amer. Math. Soc. 78, 329–332 (1955)

Zurück zum Zitat Reutenauer, C.: Free Lie algebras. University Press, Oxford (1993)MATH Reutenauer, C.: Free Lie algebras. University Press, Oxford (1993)MATH

Zurück zum Zitat Radford, D.E.: J. Algebra 58, 432–454 (1979)MathSciNetCrossRefMATH Radford, D.E.: J. Algebra 58, 432–454 (1979)MathSciNetCrossRefMATH

Zurück zum Zitat Witt, E.: Treue Darstellung Liescher Ringe. J. Reine und Angew. Math. 177, 152–160 (1937); Die Unterringe der freien Lieschen Ringe. Math. Zeitschr. 64, 195–216 (1956) Witt, E.: Treue Darstellung Liescher Ringe. J. Reine und Angew. Math. 177, 152–160 (1937); Die Unterringe der freien Lieschen Ringe. Math. Zeitschr. 64, 195–216 (1956)

Zurück zum Zitat 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) 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)

Zurück zum Zitat Maitre, D.: HPL, a mathematica implementation of the harmonic polylogarithms. Comput. Phys. Commun. 174, 222–240 (2006). [hep-ph/0507152] Maitre, D.: HPL, a mathematica implementation of the harmonic polylogarithms. Comput. Phys. Commun. 174, 222–240 (2006). [hep-ph/0507152]

Zurück zum Zitat Ablinger, J., Blümlein, J., Schneider, C.: DESY 13-064 Ablinger, J., Blümlein, J., Schneider, C.: DESY 13-064

Zurück zum Zitat Blümlein, J.: The theory of deeply inelastic scattering. Prog. Part. Nucl. Phys. 69, 28 (2013). [arXiv:1208.6087 [hep-ph]] Blümlein, J.: The theory of deeply inelastic scattering. Prog. Part. Nucl. Phys. 69, 28 (2013). [arXiv:1208.6087 [hep-ph]]

Zurück zum Zitat 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]]. 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]].

Zurück zum Zitat Euler, L.: Meditationes circa singulare serium genus. Novi Commentarii academiae scientiarum Petropolitanae 20, 140–186 (1776) Euler, L.: Meditationes circa singulare serium genus. Novi Commentarii academiae scientiarum Petropolitanae 20, 140–186 (1776)

Zurück zum Zitat 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) 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)

Zurück zum Zitat Blümlein, J., Broadhurst, D.J., Vermaseren, J.A.M.: The multiple zeta value data mine. Comput. Phys. Commun. 181, 582–625 (2010). [arXiv:0907.2557 [math-ph]] and references therein Blümlein, J., Broadhurst, D.J., Vermaseren, J.A.M.: The multiple zeta value data mine. Comput. Phys. Commun. 181, 582–625 (2010). [arXiv:0907.2557 [math-ph]] and references therein

Zurück zum Zitat Hoffman, M.E.: Hoffman’s page. http://www.usna.edu/Users/math/~meh/biblio.html Hoffman, M.E.: Hoffman’s page. http://​www.​usna.​edu/​Users/​math/​~meh/​biblio.​html

Zurück zum Zitat Fischler, S.: Irrationalité de valeurs de zéta. Sém. Bourbaki, Novembre 2002, exp. no. 910. Asterisque 294, 27–62 (2004) http://www.math.u-psud.fr/~fischler/publi.html; Colmez, P.: Arithmetique de la fonction zêta. In: Journées X-UPS 2002. La fontion zêta. Editions de l’Ecole polytechnique, pp. 37–164. Paris (2002). http://www.math.polytechnique.fr/xups/vol02.html;Waldschmidt, M.: Multiple polylogarithms: an introduction. In: Agarwal, A.K., Berndt, B.C., Krattenthaler, C.F., Mullen, G.L., Ramachandra, K., Waldschmidt, M. (eds.) Number Theory and Discrete Mathematics, pp. 1–12. Hindustan Book Agency, New Delhi (2002);Waldschmidt, M.: Valeurs zêta multiples. Une introduction. Journal de théorie des nombres de Bordeaux 12(2), 581–595 (2000);Huttner, M., Petitot, M.: Arithmeétique des fonctions d’zetas et Associateur de Drinfel’d. UFR de Mathématiques, Lille (2005)Hertling, C.: AG Mannheim-Heidelberg, SS2007;Cartier, P.: Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents. Sém. Bourbaki, Mars 2001, 53e année, exp. no. 885. Asterisque 282, 137–173 (2002) Fischler, S.: Irrationalité de valeurs de zéta. Sém. Bourbaki, Novembre 2002, exp. no. 910. Asterisque 294, 27–62 (2004) http://​www.​math.​u-psud.​fr/​~fischler/​publi.​html; Colmez, P.: Arithmetique de la fonction zêta. In: Journées X-UPS 2002. La fontion zêta. Editions de l’Ecole polytechnique, pp. 37–164. Paris (2002). http://​www.​math.​polytechnique.​fr/​xups/​vol02.​html;Waldschmidt, M.: Multiple polylogarithms: an introduction. In: Agarwal, A.K., Berndt, B.C., Krattenthaler, C.F., Mullen, G.L., Ramachandra, K., Waldschmidt, M. (eds.) Number Theory and Discrete Mathematics, pp. 1–12. Hindustan Book Agency, New Delhi (2002);Waldschmidt, M.: Valeurs zêta multiples. Une introduction. Journal de théorie des nombres de Bordeaux 12(2), 581–595 (2000);Huttner, M., Petitot, M.: Arithmeétique des fonctions d’zetas et Associateur de Drinfel’d. UFR de Mathématiques, Lille (2005)Hertling, C.: AG Mannheim-Heidelberg, SS2007;Cartier, P.: Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents. Sém. Bourbaki, Mars 2001, 53e année, exp. no. 885. Asterisque 282, 137–173 (2002)

Zurück zum Zitat 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) 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)

Zurück zum Zitat Kuipers, J., Vermaseren, J.A.M.: About a conjectured basis for multiple zeta values. [arXiv:1105.1884 [math-ph]] Kuipers, J., Vermaseren, J.A.M.: About a conjectured basis for multiple zeta values. [arXiv:1105.1884 [math-ph]]

Zurück zum Zitat 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) 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)

Zurück zum Zitat 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] 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]

Zurück zum Zitat 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). [arXiv:hep-th/9609128]MathSciNetADSCrossRefMATH 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). [arXiv:hep-th/9609128]MathSciNetADSCrossRefMATH

Zurück zum Zitat http://en.wikipedia.org/wiki/Padovan_sequence;http://www.emis.de/journals/NNJ/conferences/N2002-Padovan.html http://​en.​wikipedia.​org/​wiki/​Padovan_​sequence;http://​www.​emis.​de/​journals/​NNJ/​conferences/​N2002-Padovan.​html

Zurück zum Zitat Perrin, R.: Item 1484, L’Intermédiare des Math. 6, 76–77 (1899);Williams, A., Shanks, D.: Strong primality tests that are not sufficient. Math. Comput. 39(159), 255–300 (1982) Perrin, R.: Item 1484, L’Intermédiare des Math. 6, 76–77 (1899);Williams, A., Shanks, D.: Strong primality tests that are not sufficient. Math. Comput. 39(159), 255–300 (1982)

Zurück zum Zitat de filiis Bonaccij, L.P.: Liber abaci, Cap. 12.7. Pisa (1202);Sigler, L.E.: Fibonacci’s Liber Abaci. Springer, Berlin (2002) de filiis Bonaccij, L.P.: Liber abaci, Cap. 12.7. Pisa (1202);Sigler, L.E.: Fibonacci’s Liber Abaci. Springer, Berlin (2002)

Zurück zum Zitat Hoffman, M.E.: The algebra of multiple harmonic series. J. Algebra 194, 477–495 (1997)MathSciNetCrossRefMATH Hoffman, M.E.: The algebra of multiple harmonic series. J. Algebra 194, 477–495 (1997)MathSciNetCrossRefMATH

Zurück zum Zitat Brown, F.: Mixed tate motives over Z. Ann. Math. 175(1), 949–976 (2012)CrossRefMATH Brown, F.: Mixed tate motives over Z. Ann. Math. 175(1), 949–976 (2012)CrossRefMATH

Zurück zum Zitat The formula goes back to de Moivre, Bernoulli, Euler, and later Binet, see: Beutelspacher, A., Petri, B.: Der Goldene Schnitt. Spektrum, Heidelberg (1988) The formula goes back to de Moivre, Bernoulli, Euler, and later Binet, see: Beutelspacher, A., Petri, B.: Der Goldene Schnitt. Spektrum, Heidelberg (1988)

Zurück zum Zitat Lucas, E.: Théorie des fonctions numériques simplement périodiques. Am. J. Math. 1, 197–240 (1878)CrossRef Lucas, E.: Théorie des fonctions numériques simplement périodiques. Am. J. Math. 1, 197–240 (1878)CrossRef

Zurück zum Zitat 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)) 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))

Zurück zum Zitat Hoffman, M. E., Moen, C.: Sums of triple harmonic series. J. Number Theory 60, 329–331 (1996);Granville, A.: A decomposition of Riemann’s zeta-function. In: Motohashi, Y. (ed.) Analytic Number Theory, London Mathematical Society. Lecture Note Series, vol. 247, pp. 95–101. Cambridge University Press, Cambridge (1997); Zagier, D.: Multiple Zeta Values. (preprint) Hoffman, M. E., Moen, C.: Sums of triple harmonic series. J. Number Theory 60, 329–331 (1996);Granville, A.: A decomposition of Riemann’s zeta-function. In: Motohashi, Y. (ed.) Analytic Number Theory, London Mathematical Society. Lecture Note Series, vol. 247, pp. 95–101. Cambridge University Press, Cambridge (1997); Zagier, D.: Multiple Zeta Values. (preprint)

Zurück zum Zitat Hoffman, M.E.: Multiple harmonic series. Pac. J. Math. 152, 275–290 (1992)CrossRefMATH Hoffman, M.E.: Multiple harmonic series. Pac. J. Math. 152, 275–290 (1992)CrossRefMATH

Zurück zum Zitat 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]] 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]]

Zurück zum Zitat 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]] 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]]

Zurück zum Zitat Hoffmann, M.E., Ohno, Y.: Relations of multiple zeta values and their algebraic expressions. J. Algebra 262, 332–347 (2003)MathSciNetCrossRef Hoffmann, M.E., Ohno, Y.: Relations of multiple zeta values and their algebraic expressions. J. Algebra 262, 332–347 (2003)MathSciNetCrossRef

Zurück zum Zitat Zudilin, V.V.: Algebraic relations for multiple zeta values. Uspekhi Mat. Nauk 58(1), 3–22 Zudilin, V.V.: Algebraic relations for multiple zeta values. Uspekhi Mat. Nauk 58(1), 3–22

Zurück zum Zitat Ihara, K., Kaneko, M.: A note on relations of multiple zeta values (preprint) Ihara, K., Kaneko, M.: A note on relations of multiple zeta values (preprint)

Zurück zum Zitat 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)MathSciNetCrossRef 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)MathSciNetCrossRef

Zurück zum Zitat Ohno, Y.: A generalization of the duality and sum formulas on the multiple zeta values. J. Number Theory 74, 189–209 (1999)MathSciNetCrossRef Ohno, Y.: A generalization of the duality and sum formulas on the multiple zeta values. J. Number Theory 74, 189–209 (1999)MathSciNetCrossRef

Zurück zum Zitat Ohno, Y., Zagier, D.: Indag. Math. (N.S.) 12, 483–487 (2001) Ohno, Y., Zagier, D.: Indag. Math. (N.S.) 12, 483–487 (2001)

Zurück zum Zitat Ohno, Y., Wakabayashi, N.: Cyclic sum of multiple zeta values. Acta Arithmetica 123, 289–295 (2006)MathSciNetADSCrossRefMATH Ohno, Y., Wakabayashi, N.: Cyclic sum of multiple zeta values. Acta Arithmetica 123, 289–295 (2006)MathSciNetADSCrossRefMATH

Zurück zum Zitat 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] 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]

Zurück zum Zitat Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compositio Math. 142, 307–338 (2006); Preprint MPIM2004-100 Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compositio Math. 142, 307–338 (2006); Preprint MPIM2004-100

Zurück zum Zitat 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) 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)

Zurück zum Zitat Berndt, B.C.: Ramanujan’s Notebooks, Part IV, pp. 323–326. Springer, New York (1994) Berndt, B.C.: Ramanujan’s Notebooks, Part IV, pp. 323–326. Springer, New York (1994)

Zurück zum Zitat Ablinger, J., Blümlein, J., Raab, C., Schneider, C., Wißbrock, F.: DESY 13–063 Ablinger, J., Blümlein, J., Raab, C., Schneider, C., Wißbrock, F.: DESY 13–063

Zurück zum Zitat Weinzierl, S.: Feynman Graphs. [arXiv:1301.6918 [hep-ph]] Weinzierl, S.: Feynman Graphs. [arXiv:1301.6918 [hep-ph]]

Zurück zum Zitat 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, M.E.: Hopf Algebras. Benjamin, New York (1969) 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, M.E.: Hopf Algebras. Benjamin, New York (1969)

Zurück zum Zitat Goncharov, A.B.: Multiple polylogarithms, cyclotomy and modular complexes. Math. Res. Lett. 5, 497–516 (1998). [arXiv:1105.2076 [math.AG]] Goncharov, A.B.: Multiple polylogarithms, cyclotomy and modular complexes. Math. Res. Lett. 5, 497–516 (1998). [arXiv:1105.2076 [math.AG]]

Zurück zum Zitat Gehrmann, T., Remiddi, E.: Two loop master integrals for γ∗ → 3 jets: the planar topologies. Nucl. Phys. B 601, 248–286 (2001). [hep-ph/0008287]; Numerical evaluation of twodimensional harmonic polylogarithms. Comput. Phys. Commun. 144, 200–223 (2002). [hep-ph/0111255] Gehrmann, T., Remiddi, E.: Two loop master integrals for γ^∗ → 3 jets: the planar topologies. Nucl. Phys. B 601, 248–286 (2001). [hep-ph/0008287]; Numerical evaluation of twodimensional harmonic polylogarithms. Comput. Phys. Commun. 144, 200–223 (2002). [hep-ph/0111255]

Zurück zum Zitat Weinzierl, S.: Symbolic expansion of transcendental functions. Comput. Phys. Commun. 145, 357–370 (2002). [math-ph/0201011] Weinzierl, S.: Symbolic expansion of transcendental functions. Comput. Phys. Commun. 145, 357–370 (2002). [math-ph/0201011]

Zurück zum Zitat Moch, S.-O., Uwer, P.: XSummer: transcendental functions and symbolic summation in FORM. Comput. Phys. Commun. 174, 759–770 (2006). [math-ph/0508008] Moch, S.-O., Uwer, P.: XSummer: transcendental functions and symbolic summation in FORM. Comput. Phys. Commun. 174, 759–770 (2006). [math-ph/0508008]

Zurück zum Zitat Appell, P.: Sur Les Fonctions Hypérgéometriques de Plusieurs Variables. Gauthier-Villars, Paris (1925);Appell, P., Kampé de Fériet, J.: Fonctions Hypérgéometriques; Polynômes d’Hermite. Gauthier-Villars, Paris (1926);Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935);Erdélyi, A. (ed.): Higher Transcendental Functions, Bateman Manuscript Project, vol. I. McGraw-Hill, New York (1953);Exton, H.: Multiple Hypergeometric Functions and Applications. Ellis Horwood Limited, Chichester (1976); Handbook of Hypergeometric Integrals. Ellis Horwood Limited, Chichester (1978);Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966);Schlosser, H.: (this volume);Anastasiou, C., Glover, E.W.N., Oleari, C.: Application of the negative dimension approach to massless scalar box integrals. Nucl. Phys. B 565, 445–467 (2000). [hep-ph/9907523]; Scalar one loop integrals using the negative dimension approach. Nucl. Phys. B 572, 307–360 (2000). [hep-ph/9907494];Glover, E.W.N.: (this volume) Appell, P.: Sur Les Fonctions Hypérgéometriques de Plusieurs Variables. Gauthier-Villars, Paris (1925);Appell, P., Kampé de Fériet, J.: Fonctions Hypérgéometriques; Polynômes d’Hermite. Gauthier-Villars, Paris (1926);Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935);Erdélyi, A. (ed.): Higher Transcendental Functions, Bateman Manuscript Project, vol. I. McGraw-Hill, New York (1953);Exton, H.: Multiple Hypergeometric Functions and Applications. Ellis Horwood Limited, Chichester (1976); Handbook of Hypergeometric Integrals. Ellis Horwood Limited, Chichester (1978);Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966);Schlosser, H.: (this volume);Anastasiou, C., Glover, E.W.N., Oleari, C.: Application of the negative dimension approach to massless scalar box integrals. Nucl. Phys. B 565, 445–467 (2000). [hep-ph/9907523]; Scalar one loop integrals using the negative dimension approach. Nucl. Phys. B 572, 307–360 (2000). [hep-ph/9907494];Glover, E.W.N.: (this volume)

Zurück zum Zitat 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). [arXiv:1206.2252 [hep-ph]] 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). [arXiv:1206.2252 [hep-ph]]

Zurück zum Zitat 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]] 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]]

Zurück zum Zitat Lang, S.: Algebra, 3rd edn. Springer, New York (2002)CrossRefMATH Lang, S.: Algebra, 3rd edn. Springer, New York (2002)CrossRefMATH

Zurück zum Zitat 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 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

Zurück zum Zitat 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 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

Zurück zum Zitat 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). [hep-th/9803091]ADS 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). [hep-th/9803091]ADS

Zurück zum Zitat 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] 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]

Zurück zum Zitat Davydychev, A.I., Kalmykov, M.Y.: Massive Feynman diagrams and inverse binomial sums. Nucl. Phys. B 699, 3–64 (2004). [hep-th/0303162];Weinzierl, S.: Expansion around half integer values, binomial sums and inverse binomial sums. J. Math. Phys. 45, 2656–2673 (2004). [hep-ph/0402131];Kalmykov, M.Y.: Gauss’ hypergeometric function: Reduction, \(\varepsilon\)-expansion for integer/halfinteger parameters and Feynman diagrams. JHEP 0604, 056 (2006). [hep-th/0602028];Huber, T., Maitre, D.: Expanding hypergeometric functions about half-integer parameters. Comput. Phys. Commun. 178, 755–776 (2008). [arXiv:0708.2443 [hep-ph]] Davydychev, A.I., Kalmykov, M.Y.: Massive Feynman diagrams and inverse binomial sums. Nucl. Phys. B 699, 3–64 (2004). [hep-th/0303162];Weinzierl, S.: Expansion around half integer values, binomial sums and inverse binomial sums. J. Math. Phys. 45, 2656–2673 (2004). [hep-ph/0402131];Kalmykov, M.Y.: Gauss’ hypergeometric function: Reduction, \(\varepsilon\)-expansion for integer/halfinteger parameters and Feynman diagrams. JHEP 0604, 056 (2006). [hep-th/0602028];Huber, T., Maitre, D.: Expanding hypergeometric functions about half-integer parameters. Comput. Phys. Commun. 178, 755–776 (2008). [arXiv:0708.2443 [hep-ph]]

Zurück zum Zitat Ablinger, J., Blümlein, J., Raab, C., Schneider, C.: (in preparation) Ablinger, J., Blümlein, J., Raab, C., Schneider, C.: (in preparation)

Zurück zum Zitat 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]] 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]]

Zurück zum Zitat 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) 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)

Zurück zum Zitat 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). [arXiv:1104.3449 [hep-ph]] 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). [arXiv:1104.3449 [hep-ph]]

Zurück zum Zitat Blümlein, J., Vogt, A.: The evolution of unpolarized singlet structure functions at small x. Phys. Rev. D 58, 014020 (1998). [hep-ph/9712546] Blümlein, J., Vogt, A.: The evolution of unpolarized singlet structure functions at small x. Phys. Rev. D 58, 014020 (1998). [hep-ph/9712546]

Zurück zum Zitat Nielsen, N.: Handbuch der Theorie der Gammafunktion. Teubner, Leipzig (1906); Reprinted by Chelsea Publishing Company, Bronx, New York (1965) Nielsen, N.: Handbuch der Theorie der Gammafunktion. Teubner, Leipzig (1906); Reprinted by Chelsea Publishing Company, Bronx, New York (1965)

Zurück zum Zitat Landau, E.: Über die Grundlagen der Theorie der Fakultätenreihen, S.-Ber. math.-naturw. Kl. Bayerische Akad. Wiss. München, 36, 151–218 (1906) Landau, E.: Über die Grundlagen der Theorie der Fakultätenreihen, S.-Ber. math.-naturw. Kl. Bayerische Akad. Wiss. München, 36, 151–218 (1906)

Zurück zum Zitat Carlson, F.D.: Sur une classe de séries de Taylor. PhD thesis, Uppsala University (1914);see also http://en.wikipedia.org/wiki/Carlson/%27s_theorem Carlson, F.D.: Sur une classe de séries de Taylor. PhD thesis, Uppsala University (1914);see also http://​en.​wikipedia.​org/​wiki/​Carlson/​%27s_​theorem

Zurück zum Zitat 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] 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]

Zurück zum Zitat Kotikov, A.V., Velizhanin, V.N.: Analytic continuation of the Mellin moments of deep inelastic structure functions. [hep-ph/0501274] Kotikov, A.V., Velizhanin, V.N.: Analytic continuation of the Mellin moments of deep inelastic structure functions. [hep-ph/0501274]

Zurück zum Zitat Blümlein, J., Riemann, T., Schneider, C.: DESY Annual Report (2013) Blümlein, J., Riemann, T., Schneider, C.: DESY Annual Report (2013)