Structural relations of harmonic sums and Mellin transforms up to weight w = 5

Computer Physics Communications

Volumn 180, Issue 11, 2009, Pages 2218-2249

  Blümlein, Johannes ^a  

^a DESY   (Germany)

Author keywords

Analytic continuation; Asymptotic expansions; Differentiation; Factorial series; Harmonic sums; Structural relations

Indexed keywords

DIFFERENTIATION (CALCULUS); HARMONIC ANALYSIS; SCALES (WEIGHING INSTRUMENTS);

ANALYTIC CONTINUATION; ASYMPTOTIC EXPANSION; FACTORIAL SERIES; HARMONIC SUMS; STRUCTURAL RELATIONS;

HARMONIC FUNCTIONS;

EID: 70149123628     PISSN: 00104655     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cpc.2009.07.004     Document Type: Article

References (148)

1. Gonzalez-Arroyo A., Lopez C., and Yndurain F.J. Nucl. Phys. B 153 (1979) 161
2. Gonzalez-Arroyo A., and Lopez C. Nucl. Phys. B 166 (1980) 429
3. Blümlein J., and Kurth S. DESY 97-160. Phys. Rev. D 60 (1999) 014018. hep-ph/9708388
4. Vermaseren J.A.M. Int. J. Modern Phys. A 14 (1999) 2037
5. Nielsen N. Der Eulersche Dilogarithmus und seine Verallgemeinerungen. Nova Acta Leopoldina XC 3 (1909)
6. Kölbig K.S. SIAM J. Math. Anal. 16 (1986) 1232
7. Zijlstra E.B., and van Neerven W.L. Phys. Lett. B 273 (1991) 476
8. Zijlstra E.B., and van Neerven W.L. Nucl. Phys. B 383 (1992) 525
9. van Neerven W.L., and Zijlstra E.B. Phys. Lett. B 272 (1991) 127
10. Kazakov D.I., and Kotikov A.V. Nucl. Phys. B 307 (1988) 721 Erratum
11. Kazakov D.I., and Kotikov A.V. Nucl. Phys. B 345 (1990) 299
12. Sanchez Guillen J., Miramontes J., Miramontes M., Parente G., and Sampayo O.A. Nucl. Phys. B 353 (1991) 337
13. Moch S., and Vermaseren J.A.M. Nucl. Phys. B 573 (2000) 853. arXiv:hep-ph/9912355
14. Larin S.A., van Ritbergen T., and Vermaseren J.A.M. Nucl. Phys. B 427 (1994) 41
15. Larin S.A., Nogueira P., van Ritbergen T., and Vermaseren J.A.M. Nucl. Phys. B 492 (1997) 338. arXiv:hep-ph/9605317
16. Retey A., and Vermaseren J.A.M. Nucl. Phys. B 604 (2001) 281. arXiv:hep-ph/0007294
17. Blümlein J., and Vermaseren J.A.M. Phys. Lett. B 606 (2005) 130. arXiv:hep-ph/0411111
18. Moch S., Vermaseren J.A.M., and Vogt A. Nucl. Phys. B 688 (2004) 101. arXiv:hep-ph/0403192
19. Vogt A., Moch S., and Vermaseren J.A.M. Nucl. Phys. B 691 (2004) 129. arXiv:hep-ph/0404111
20. Bierenbaum I., Blümlein J., and Klein S. Nucl. Phys. Proc. Suppl. 183 (2008) 162. arXiv:0812.2427 [hep-ph]
21. J. Blümlein, M. Kauers, S. Klein, C. Schneider, Inverting exact functions from Mellin moments in quantum field theory, DESY 09-002
22. Buza M., Matiounine Y., Smith J., Migneron R., and van Neerven W.L. Nucl. Phys. B 472 (1996) 611. arXiv:hep-ph/9601302
23. Buza M., Matiounine Y., Smith J., and van Neerven W.L. Nucl. Phys. B 485 (1997) 420. arXiv:hep-ph/9608342
24. Blümlein J., De Freitas A., van Neerven W.L., and Klein S. Nucl. Phys. B 755 (2006) 272. arXiv:hep-ph/0608024
25. Bierenbaum I., Blümlein J., and Klein S. Phys. Lett. B 648 (2007) 195. arXiv:hep-ph/0702265
26. Bierenbaum I., Blümlein J., and Klein S. Acta Phys. Polon. B 39 (2008) 1531. arXiv:0901.0669 [hep-ph]
27. Bierenbaum I., Blümlein J., and Klein S. Nucl. Phys. B 780 (2007) 40. arXiv:hep-ph/0703285
28. Bierenbaum I., Blümlein J., Klein S., and Schneider C. Nucl. Phys. B 803 (2008) 1. arXiv:0803.0273 [hep-ph]
29. Matsuura T., van der Marck S.C., and van Neerven W.L. Nucl. Phys. B 319 (1989) 570
30. Hamberg R., van Neerven W.L., and Matsuura T. Nucl. Phys. B 359 (1991) 343 Erratum
31. Hamberg R., van Neerven W.L., and Matsuura T. Nucl. Phys. B 644 (2002) 403
32. Catani S., de Florian D., and Grazzini M. JHEP 0105 (2001) 025. arXiv:hep-ph/0102227
33. Harlander R.V., and Kilgore W.B. Phys. Rev. D 64 (2001) 013015. arXiv:hep-ph/0102241
34. Harlander R.V., and Kilgore W.B. Phys. Rev. Lett. 88 (2002) 201801. arXiv:hep-ph/0201206
35. Anastasiou C., and Melnikov K. Nucl. Phys. B 646 (2002) 220. arXiv:hep-ph/0207004
36. Ravindran V., Smith J., and van Neerven W.L. Nucl. Phys. B 665 (2003) 325. arXiv:hep-ph/0302135
37. Ravindran V., Smith J., and van Neerven W.L. Nucl. Phys. B 704 (2005) 332. arXiv:hep-ph/0408315
38. Rijken P.J., and van Neerven W.L. Phys. Lett. B 386 (1996) 422. arXiv:hep-ph/9604436
39. Rijken P.J., and van Neerven W.L. Nucl. Phys. B 487 (1997) 233. arXiv:hep-ph/9609377
40. Rijken P.J., and van Neerven W.L. Phys. Lett. B 392 (1997) 207. arXiv:hep-ph/9609379
41. Blümlein J., Ravindran V., and van Neerven W.L. Acta Phys. Polon. B 29 (1998) 2581. arXiv:hep-ph/9806355
42. Blümlein J., Ravindran V., and van Neerven W.L. Nucl. Phys. B 586 (2000) 349. arXiv:hep-ph/0004172
43. Mitov A., and Moch S.O. Nucl. Phys. B 751 (2006) 18. arXiv:hep-ph/0604160
44. Penin A.A. Phys. Rev. Lett. 95 (2005) 010408. arXiv:hep-ph/0501120
45. Becher T., and Melnikov K. JHEP 0706 (2007) 084. arXiv:0704.3582 [hep-ph]
46. Czakon M., Gluza J., and Riemann T. Nucl. Phys. B 751 (2006) 1. arXiv:hep-ph/0604101
47. Czakon M., Gluza J., and Riemann T. Phys. Rev. D 71 (2005) 073009. arXiv:hep-ph/0412164
48. Blümlein J., and Ravindran V. Nucl. Phys. B 716 (2005) 128. arXiv:hep-ph/0501178
49. Blümlein J., and Ravindran V. Nucl. Phys. B 749 (2006) 1. arXiv:hep-ph/0604019
50. J. Blümlein, S. Moch, in preparation
51. Blümlein J., and Klein S. arXiv:0706.2426 [hep-ph]
52. Poincaré H. Acta Math. 4 (1884) 201
53. Lappo-Danilevsky J.A. Mémoirs sur la Théorie des Systèmes Différentielles Linéaires (1953), Chelsea Publ. Co., New York, NY
54. Chen K.T. Trans. Amer. Math. Soc. 156 3 (1971) 359
55. Remiddi E., and Vermaseren J.A.M. Int. J. Modern Phys. A 15 (2000) 725. arXiv:hep-ph/9905237
56. Hoang Ngoc Minh, Petitot M., and van der Hoeven J. Discrete Math. 225 (2000) 217
57. Schwartz L. Théorie des Distributions, vols. I and II (1951), Hermann and Cie, Paris
58. Vladimirov V.S. Gleichungen der Mathematischen Physik (1972), DVW, Berlin
59. Yoshida K. Functional Analysis (1978), Springer, Berlin
60. Blümlein J. Comput. Phys. Comm. 159 (2004) 19. arXiv:hep-ph/0311046
61. Hoffman M.E. J. Algebra 194 (1997) 477
62. Hoffman M.E. Nucl. Phys. (Proc. Suppl.) 135 (2004) 215. arXiv:math/0406589
63. Witt E. J. Reine Angew. Math. 177 (1937) 152
64. Witt E. Math. Z. 64 (1956) 195
65. Reutenauer C. Free Lie Algebras (1993), Clarendon Press, Oxford
66. See e.g. Hardy G.H., and Wright E.M. An Introduction to the Theory of Numbers (2002), Clarendon Press, Oxford
67. Lyndon R.C. Trans. Amer. Math. Soc. 77 (1954) 202
68. Lyndon R.C. Trans. Amer. Math. Soc. 78 (1955) 329
69. Radford D.E. J. Algebra 58 (1979) 432
70. Euler L. Novi Comm. Acad. Sci. Petropol. 20 (1775) 140
71. Zagier D. First European Congress of Mathematics, vol. II (1994), Birkhäuser, Boston p. 497
72. Waldschmidt M. J. Théor. Nombres Bordeaux 12 (2000) 581. http://www.math.jussieu.fr/~miw/articles/ps/mpl.ps Multiple polylogarithms, Lectures at the Institute of Math. Sciences (Chennai, Nov. 2000); update version
73. Borwein J.M., Bradley D.M., Broadhurst D.J., and Lisonek P. Trans. Amer. Math. Soc. 353 (2001) 907. arXiv:math/9910045
74. Blümlein J., Broadhurst D., and Vermaseren J.A.M. The multiple zeta value data mine. DESY 09-003. arXiv:0907.2557 [math-ph]
75. Goncharov A.B. Math. Res. Lett. 5 (1998) 497
76. Moch S., Uwer P., and Weinzierl S. J. Math. Phys. 43 (2002) 3363. arXiv:hep-ph/0110083
77. Nielsen N. Handbuch der Theorie der Gammafunktion (1906), Teubner, Leipzig reprinted by
78. Nielsen N. Handbuch der Theorie der Gammafunktion (1965), Chelsea Publishing Company, Bronx, NY
79. Landau E. Über die Grundlagen der Theorie der Fakultätenreihen. S.-Ber. math.-naturw. Kl. Bayerische Akad. Wiss. München 36 (1906) 151-218
80. K. Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, 1947
81. Blümlein J. Nucl. Phys. (Proc. Suppl.) 135 (2004) 225. arXiv:hep-ph/0407044
82. Blümlein J. Comput. Phys. Comm. 133 (2000) 76. arXiv:hep-ph/0003100
83. Blümlein J., and Moch S.O. Phys. Lett. B 614 (2005) 53. arXiv:hep-ph/0503188
84. Hastings Jr. C. Approximations for Digital Computers (1953), Princeton University Press, Princeton, NJ
85. Abramowitz M., and Stegun I.A. Handbook of Mathematical Functions (1964), NBS, Washington
86. Lyusternik L.A., Chervonenkis O.A., and Yanpol'skii A.R. Handbook for Computing of Elementary Functions (1963), Fizmatgiz, Moscow Russian ed.:
87. Lyusternik L.A., Chervonenkis O.A., and Yanpol'skii A.R. Handbook for Computing of Elementary Functions (1964), Pergamon Press, New York
88. Lanczos C. J. Math. Phys. 17 (1938) 123
89. Lösch F., and Schoblik F. Die Fakultät und verwandte Funktionen (1951), Teubner, Stuttgart
90. Alekhin S.I., and Blümlein J. Phys. Lett. B 594 (2004) 299. arXiv:hep-ph/0404034
91. See also:. Kotikov A.V., and Velizhanin V.N. arXiv:hep-ph/0501274
92. Graudenz D., Hampel M., Vogt A., and Berger C. Z. Phys. C 70 (1996) 77. arXiv:hep-ph/9506333
93. Blümlein J. arXiv:0901.0837 [math-ph]
94. Berends F.A., van Neerven W.L., and Burgers G.J.H. Nucl. Phys. B 297 (1988) 429 Erratum
95. Berends F.A., van Neerven W.L., and Burgers G.J.H. Nucl. Phys. B 304 (1988) 921
96. Blümlein J., and Kawamura H. Phys. Lett. B 553 (2003) 242. arXiv:hep-ph/0211191
97. Blümlein J., De Freitas A., and van Neerven W. PoS RADCOR2007 (2007) 005. arXiv:0812.1588 [hep-ph]
98. Wilson K.G. Phys. Rev. 179 (1969) 1499
99. Brandt R.A. Fortsch. Phys. 18 (1970) 249
100. Frishman Y. Ann. Physics 66 (1971) 373
101. Brandt R.A., and Preparata G. Fortsch. Phys. 20 (1972) 571
102. Mueller A.H. Phys. Rep. 73 (1981) 237 and references therein
103. Blümlein J., and Kochelev N. Nucl. Phys. B 498 (1997) 285. arXiv:hep-ph/9612318
104. Broadhurst D.J. Eur. Phys. J. C 8 (1999) 311. arXiv:hep-th/9803091
105. Brown F. Commun. Math. Phys. 287 (2009) 925. arXiv:0804.1660 [math.AG]
106. Y. Andre, in: Proceedings of the Workshop "Motives, Quantum Field Theory, and Pseudodifferential Operators", held at the Clay Mathematics Institute, Boston University, June 2-14, 2008, submitted for publication
107. Barnes E.W. Proc. London Math. Soc. (2) 6 (1908) 141
108. Barnes E.W. Quart. J. Math. 41 (1910) 136
109. Mellin H. Math. Ann. 68 (1910) 305
110. Whittaker E.T., and Watson G.N. A Course of Modern Analysis (1927), Cambridge University Press, Cambridge (reprinted 1996)
111. Titchmarsh E.C. Introduction to the Theory of Fourier Integrals (1937), Oxford, Clarendon Press (2nd edition 1948)
112. Bailey W.N. Generalized Hypergeometric Series (1935), Cambridge University Press, Cambridge
113. Slater L.J. Generalized Hypergeometric Functions (1966), Cambridge University Press, Cambridge
114. Kalmykov M.Y., Kniehl B.A., Ward B.F.L., and Yost S.A. arXiv:0810.3238 [hep-th]
115. Kalmykov M.Y., and Kniehl B.A. arXiv:0807.0567 [hep-th]
116. Kalmykov M.Y., Ward B.F.L., and Yost S.A. JHEP 0711 (2007) 009. arXiv:0708.0803 [hep-th]
117. Vermaseren J.A.M., Vogt A., and Moch S. Nucl. Phys. B 724 (2005) 3. arXiv:hep-ph/0504242
118. Schneider C. Ann. Comb. 9 1 (2005) 75
119. Schneider C. Proc. ISSAC'05 (2005), ACM Press 285
120. Schneider C. Proc. FPSAC'07 (2007) 1
121. Schneider C. J. Differ. Equations Appl. 11 9 (2005) 799
122. Schneider C. J. Algebra Appl. 6 3 (2007) 415
123. Schneider C. Sém. Lothar. Combin. 56 (2007) Article B56b and Habilitation Thesis, JKU Linz, 2007
124. E. Carlson, Thesis, Uppsala, 1914
125. Titchmarsh E.C. Theory of Functions (1939), Oxford University Press, Oxford (Ch. 9.5)
126. Lidl R., and Niederreiter H. Finite Fields. Encyclopedia of Mathematics vol. 20 (1983), Addison-Wesley, Reading, MA
127. Lang S. Algebra (2002), Springer, Berlin
128. Weinzierl S. J. Math. Phys. 45 (2004) 2656. arXiv:0705.0900 [hep-ph]
129. Dittmar M., et al. arXiv:hep-ph/0511119
130. Alekhin S., et al. arXiv:hep-ph/0601013
131. Sloane N.J.A. The on-line encyclopedia of integer sequences. http://www.research.att.com/~njas/sequences/ sequence:A001333
132. Lu T. Discrete Math. 154 (1996) 153
133. A.M. Legendre, Exercises sur le calcul intégral, tome II, Paris, 1817, p. 45
134. J. Stirling, Methodus differentialis sive tractatus de summatione et interpolatione serierum infinitarum, London, 1730, p. 27
135. P. Cartier, private communication, June 2008
136. Hurwitz A. Z. Math. Physik 27 (1882) 86
137. Hasse H. Math. Z. 32 (1930) 458
138. Euler L. Novi Comm. Acad. Sci. Petropolitanae 1 (1775) 140
139. C. Costermans, J.N. Enjalbert, Hoang Ngoc Minh, M. Petitot, in: Proceedings ISSAC '05, Beijing, China, p. 100
140. Blümlein J. KITP Workshop on Collider Physics, Jan. 12-Apr. 2, 2004, Santa Barbara, CA. http://online.kitp.ucsb.edu/online/collider04/bluemlein/ Transparencies presented at the Lille MZV2005 Workshop, June 2005
141. Nielsen N. Traité Élémentaire des Nombres de Bernoulli (1923), Gauthier-Villars et Cie, Paris
142. Saalschütz L. Vorlesungen über Bernoullische Zahlen (1893), Berlin
143. Weisstein E.W. CRC Concise Encyclopedia of Mathematics (1998), CRC Press, London
144. Flajolet Ph., Gourdon X., and Dumas P. Theoret. Comput. Sci. 144 (1995) 3
145. Devoto A., and Duke D.W. Riv. Nuovo Cimento 7 (1981) 1
146. Blümlein J., and Kawamura H. Nucl. Phys. B 708 (2005) 467. arXiv:hep-ph/0409289
147. Blümlein J., and Kawamura H. Eur. Phys. J. C 51 (2007) 317. arXiv:hep-ph/0701019
148. Levine M.J., Remiddi E., and Roskies R. Phys. Rev. D 20 (1979) 2068