REFERENCES ON MULTIPLE ZETA VALUES AND EULER SUMS

Compiled by Michael Hoffman

The list is in (approximate) chronological order within each category. While the categorization of some works is a bit arbitrary, I have generally tried to place each one in the most inclusive category that seemed appropriate.

This list is revised regularly. Report errors and omissions to meh@usna.edu.

Author index

A. DOUBLE HARMONIC SERIES

  1. P. H. Fuss (ed.), Correspondance Mathématique et Physique de quelques célèbres Géomètres (Tome 1), St. Petersburg, 1843.
  2. L. Euler, `Meditationes circa singulare serierum genus,' Novi Comm. Acad. Sci. Petropol. 20 (1775), 140-186. Reprinted in Opera Omnia, ser. I, vol. 15, B. G. Teubner, Berlin, 1927, pp. 217-267.
  3. N. Nielsen, Die Gammafunktion, Chelsea, New York, 1965. Reprint of Handbuch der Theorie der Gammafunktion (1906) and Theorie der Integrallogarithmus und verwandter Transzendenten (1906).
  4. L. Tornheim, `Harmonic double series,' Amer. J. Math. 72(1950), 303-314.
  5. G. T. Williams, `A new method of evaluating ζ(2n),' Amer. Math. Monthly 60 (1953), 19-25.
  6. T. M. Apostol and T. H. Vu, `Dirichlet series related to the Riemann zeta function,' J. Number Theory 19 (1982), 85-102.
  7. M. V. Subbarao and R. Sitaramachandrarao, `On some infinite series of L. J. Mordell and their analogues', Pacific J. Math. 119 (1985), 245-255.
  8. R. E. Crandall and J. P. Buhler, `On the evaluation of Euler sums,' Experiment. Math. 3 (1994), 275-285.
  9. D. Borwein and J. M. Borwein, `On an intriguing integral and some series related to ζ(4),' Proc. Amer. Math. Soc. 123 (1995), 1191-1198.
  10. J. G. Huard, K. S. Williams, and Zhang Nan-Yue, `On Tornheim's double series,' Acta Arithmetica 75 (1996), 105-117.
  11. M-A. Coppo, `Sur les sommes d'Euler divergentes,' Expositiones Mathematicae 18 (2000), 297-308.
  12. Chu Wenchang, `Symmetric functions and the Riemann zeta series,' Indian J. Pure Appl. Math. 31 (2000), 1677-1689.
  13. K. Boyadzhiev, `Evaluation of Euler-Zagier sums,' Internat. J. Math. Math. Sci. 27 (2001), 407-412.
  14. K. Boyadzhiev, `Consecutive evaluation of Euler sums,' Internat. J. Math. Math. Sci. 29 (2002), 555-561.
  15. H. Tsumura, `On some combinatorial relations for Tornheim's double series,' Acta Arithmetica 105 (2002), 239-252.
  16. T. M. Rassias and H. M. Srivastava, `Some classes of infinite series associated with the Riemann zeta and polygamma functions and generalized harmonic numbers,' Appl. Math. and Comp. 131 (2002), 593-605.
  17. M. W. Coffey, `On some log-cosine integrals related to ζ(3), ζ(4), and ζ(6),' J. Comp. Appl. Math. 153 (2003), 205-215.
  18. H. Tsumura, `On alternating analogues of Tornheim's double series,' Proc. Amer. Math. Soc. 131 (2003), 3633-3641.
  19. H. Tsumura, `Evaluation formulas for Tornheim's type of alternating double series,' Math. Comp. 73 (2004), 251-258.
  20. H. Tsumura, `On evaluation formulas for double L-values,' Bull. Austral. Math. Soc. 70 (2004), 213-221.
  21. D. Terhune, `Evaluation of double L-values,' J. Number Theory 105 (2004), 275-301.
  22. R. Masri, `The Herglotz-Zagier function, double zeta values, and values of L-series,' J. Number Theory 106 (2004), 219-237.
  23. M. W. Coffey, `On one-dimensional digamma and polygamma series related to the evaluation of Feynman diagrams,' J. Comp. Appl. Math. 183 (2005), 84-100.
  24. O. Espinosa and V. H. Moll, `The evaluation of Tornheim double sums, Part I,' J. Number Theory 116 (2006), 200-229; preprint CA/0505647.
  25. K-W. Chen and M. Eie, `Explicit evaluations of extended Euler sums,' J. Number Theory 117 (2006), 31-52.
  26. D. M. Bradley, `A q-analog of Euler's decomposition formula for the double zeta function,' Int. J. Math. Math. Sci. 2006 (2006), 3453-3458.
  27. H. Tsumura, `On some functional relations between Mordell-Tornheim double L-functions and Dirichlet L-functions,' J. Number Theory 120 (2006), 161-178.
  28. H. Gangl, M. Kaneko, D. Zagier, `Double zeta values and modular forms,' in Automorphic forms and zeta functions, S. Böcherer et. al. (eds.), World Scientific, Hackensack, NJ, 2006, pp. 71-106; preprint MPIM2005-96.
  29. T. Nakamura, `A functional relation for the Tornheim zeta function,' Acta Arithetica 125 (2006), 257-263.
  30. H. Tsumura, `On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function,' Math. Proc. Camb. Philos. Soc. 142 (2007), 395-405.
  31. J. M. Borwein, `Hilbert's inequality and Witten's zeta-function," Amer. Math. Monthly 115 (2008), 125-137.
  32. M. W. Coffey, `On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions,' J. Math. Phys. 49 (2008), 043510.
  33. X. Zhou, T. Cai, and D. Bradley, `Signed q-analogs of Tornheim's double series,' Proc. Amer. Math. Soc. 136 (2008), 2689-2698.
  34. T. Machide, `Generators for vector spaces consisting of double zeta values with even weight,' preprint 0802.1565[NT].

B. TRIPLE HARMONIC SERIES

  1. R. Sitaramachandrarao and M. V. Subbarao, `Transformation formulae for multiple series,' Pacific J. Math. 113 (1984), 471-479.
  2. C. Markett, `Triple sums and the Riemann zeta function,' J. Number Theory 48 (1994), 113-132.
  3. J. M. Borwein and R. Girgensohn, `Evaluation of triple Euler sums,' with appendix `Euler sums in quantum field theory' by D. J. Broadhurst, Electronic J. Combinatorics 3 (1996), R23.
  4. M. E. Hoffman and C. Moen, `Sums of triple harmonic series,' J. Number Theory 60 (1996), 329-331.
  5. A. Panholzer and H. Prodinger, `Computer-free evaluation of an infinite double sum via Euler sums,' Sém. Lothar. Combin. 55 (2005), art. B55a.
  6. K. Matsumoto, T. Nakamura, and H. Tsumura, `Functional relations and special values of Mordell-Tornheim triple zeta and L-functions,' Proc. Amer. Math. Soc. 136 (2008), 2135-2145.

C. MULTIPLE HARMONIC SERIES/MULTIPLE ZETA VALUES

  1. M. E. Hoffman, `Multiple harmonic series,' Pacific J. Math. 152 (1992), 275-290.
  2. D. Zagier, `Values of zeta functions and their applications,' in First European Congress of Mathematics (Paris, 1992), Vol. II, A. Joseph et. al. (eds.), Birkhäuser, Basel, 1994, pp. 497-512.
  3. T. Q. T. Le and J. Murakami, `Kontsevich's integral for the Homfly polynomial and relations between values of the multiple zeta functions,' Topology Appl. 62 (1995), 193-206.
  4. T. Q. T. Le and J. Murakami, `Kontsevich's integral for the Kauffman polynomial,' Nagoya Math. J. 142 (1996), 39-65.
  5. A. Granville, `A decomposition of Riemann's zeta-function,' in Analytic Number Theory, London Mathematical Society Lecture Note Series 247, Y. Motohashi (ed.), Cambridge University Press, 1997, pp. 95-101.
  6. D. J. Broadhurst and D. Kreimer, `Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops,' Physics Lett. B 393 (1997), 403-412.
  7. M. E. Hoffman, `The algebra of multiple harmonic series,' J. Algebra 194 (1997), 477-495.
  8. R. E. Crandall, `Fast evaluation of multiple zeta sums,' Math. Comp. 67 (1998), 1163-1172.
  9. J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, `Combinatorial aspects of multiple zeta values,' Electronic J. Combinatorics 5 (1998), R38.
  10. Hoang Ngoc Minh, M. Petitot, and J. Van Der Hoven, `Computation of the Monodromy of Generalized Polylogarithms,' Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation (Rostock), 276-283, ACM, New York, 1998.
  11. Y. Ohno, `A generalization of the duality and sum formulas on the multiple zeta values,' J. Number Theory 74 (1999), 39-43.
  12. T. Arakawa and M. Kaneko, `Multiple zeta values, poly-Bernoulli numbers, and related zeta functions,' Nagoya Math. J. 153 (1999), 189-209.
  13. T. Takamuki, `The Kontsevich invariant and relations of multiple zeta values,' Kobe J. Math. 16 (1999), 27-43.
  14. Hoang Ngoc Minh, G. Jacob, M. Petitot, and N. E. Oussous, `Aspects combinatoires des polylogarithms et des sommes d'Euler-Zagier,' Sém. Lothar. Combin. 43 (1999), art. B43e.
  15. J. Zhao, `Analytic continuation of multiple zeta functions,' Proc. Amer. Math. Soc. 128 (2000), 1275-1283.
  16. Hoang Ngoc Minh and M. Petitot, `Lyndon words, polylogarithms, and the Riemann ζ function,' Discrete Math. 217 (2000), 273-292.
  17. M. Waldschmidt, `Valeurs zêta multiples. Une introduction,' J. Théor. Nombres Bordeaux 12 (2000), 581-595.
  18. M. Kontsevich and D. Zagier, `Periods,' in Mathematics Unlimited--2001 and Beyond, Springer, Berlin, 2001, pp. 771-808.
  19. S. Akiyama, S. Egami, and Y. Tanigawa, `Analytic continuation of multiple zeta-functions and their values at non-positive integers,' Acta Arithmetica 98 (2001), 107-116.
  20. K. Ihara and T. Takamuki, `The quantum g2 invariant and relations of multiple zeta values,' J. Knot Theory Ramifications 10 (2001), 983-997.
  21. S. Akiyama and Y. Tanigawa, `Multiple zeta values at non-positive integers,' Ramanujan J. 5 (2001), 327-351.
  22. Y. Ohno and D. Zagier, `Multiple zeta values of fixed weight, depth, and height,' Indag. Math. (N. S.) 12 (2001), 483-487.
  23. D. Bowman and D. M. Bradley, `The algebra and combinatorics of shuffles and multiple zeta values,' J. Combin. Theory Ser. A 97 (2002), 43-61.
  24. M. E. Hoffman, `Periods of mirrors and multiple zeta values,' Proc. Amer. Math. Soc. 130 (2002), 971-974.
  25. T. Terasoma, `Selberg integrals and multiple zeta values,' Compositio Math. 133 (2002), 1-24.
  26. P. Cartier, `Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents,' Astérisque 282 (2002), 137-173 (Sém. Bourbaki no. 885).
  27. U. Müller and C. Schubert, `A quantum field theoretical representation of Euler-Zagier sums,' Internat. J. Math. Math. Sci. 31 (2002), 127-148; preprint QA/9908067.
  28. T. Terasoma, `Mixed Tate motives and multiple zeta values,' Invent. Math. 149 (2002), 339-369; preprint AG/0104231.
  29. S. Fischler, `Formes linéaires en polyzêtas et intégrales multiples,' C. R. Acad. Sci. Paris, Ser. I 335 (2002), 1-4.
  30. K. Matsumoto, `On the analytic continuation of various multiple-zeta functions,' in Number Theory for the Millennium (Urbana, 2000), Vol. II, M. A. Bennett et. al. (eds.), A. K. Peters, Natick, MA, 2002, pp. 417-440.
  31. D. Bowman, D. M. Bradley, and J. H. Ryoo, `Some multi-set inclusions associated with shuffle convolutions and multiple zeta values,' European J. Combin. 24 (2003), 121-127.
  32. M. E. Hoffman and Y. Ohno, `Relations of multiple zeta values and their algebraic expression,' J. Algebra 262 (2003), 332-347; preprint QA/0010140.
  33. W. Zudilin, `Algebraic relations for multiple zeta values,' (Russian), Uspekhi Mat. Nauk 58 (2003), 3-32; English translation in Russian Math. Surveys 58 (2003), 1-29; preprint.
  34. H. Ishikawa and K. Matsumoto, `On the estimation of the order of Euler-Zagier multiple zeta-functions,' Illinois J. Math. 47 (2003), 1151-1166.
  35. M. Espie, J-C. Novelli, and G. Racinet, `Formal computations about multiple zeta values,' in From Combinatorics to Dynamical Systems (Strasbourg, 2002), IRMA Lect. Math. Theor. Phys. 3, F. Fauvet and C. Mitschi (eds.), de Gruyter, Berlin, 2003, pp. 1-16.
  36. D. Bowman and D. M. Bradley, `Resolution of some open problems concerning multiple zeta evaluations of arbitrary depth,' Compositio Math. 139 (2003), 85-100.
  37. H. Furusho, `The multiple zeta value algebra and the stable derivation algebra,' Publ. Res. Inst. Math. Sci. 39 (2003), 695-720; preprint NT/0011261.
  38. H. Furusho, `p-Adic multiple zeta values I--p-adic multiple polylogarithms and the p-adic KZ equation,' Invent. Math. 155 (2004), 253-286; preprint NT/0304085.
  39. H. Tsumura, `Combinatorial relations for Euler-Zagier sums,' Acta Arithmetica 111 (2004), 27-42.
  40. H. Tsumura, `Multiple harmonic series related to multiple Euler numbers,' J. Number Theory 106 (2004), 155-168.
  41. J. Okuda and K. Ueno, `Relations for multiple zeta values and Mellin transforms of multiple polylogarithms,' Publ. Res. Inst. Math. Sci. 40 (2004), 537-564; preprint NT/0301277.
  42. S. Ünver, `p-Adic multi-zeta values,' J. Number Theory 108 (2004), 111-156.
  43. J. Écalle, `Recent advances in the analysis of divergence and singularities,' in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations (Montreal, 2002), Y. Ilyashenko et. al. (eds.), Kluwer, Dordrecht, 2004, pp. 87-186.
  44. J. Écalle, `Multizetas, perinomal numbers, arithmetical dimorphy, and ARI/GARI,' Ann. Fac. Sci. Toulouse 13 (2004), 683-708.
  45. A. B. Goncharov and Yu. I. Manin, `Multiple ζ-motives and moduli spaces M0,n,' Compositio Math. 140 (2004), 1-14; preprint AG/0204102.
  46. D. M. Bradley, `Multiple q-zeta values,' J. Algebra 283 (2005), 752-798; preprint QA/0402093.
  47. H. Tsumura, `On Mordell-Tornheim zeta values,' Proc. Amer. Math. Soc. 133 (2005), 2387-2393.
  48. T. Nakamura, `Bernoulli numbers and multiple zeta values,' Proc. Japan Acad. Ser. A 81 (2005), 21-22.
  49. D. M. Bradley, `Partition identities for the multiple zeta function,' in Zeta Functions, Topology and Quantum Physics, Developments in Mathematics 14, T. Aoki et. al. (eds.), Springer, New York, 2005, pp. 19-29; preprint CO/0402091.
  50. M. E. Hoffman, `Algebraic aspects of multiple zeta values,' in Zeta Functions, Topology and Quantum Physics, Developments in Mathematics 14, T. Aoki et. al. (eds.), Springer, New York, 2005, pp. 51-74; preprint QA/0309425.
  51. Y. Ohno, `Sum relations for multiple zeta values,' in Zeta Functions, Topology and Quantum Physics, Developments in Mathematics 14, T. Aoki et. al. (eds.), Springer, New York, 2005, pp. 131-144.
  52. J. Okuda and K. Ueno, `The sum formula for multiple zeta values and connection problem of the formal Knizhnik-Zamolodchikov equation,' in Zeta Functions, Topology and Quantum Physics, Developments in Mathematics 14, T. Aoki et. al. (eds.), Springer, New York, 2005, pp. 145-170; preprint NT/0310259.
  53. M. Waldschmidt, `Hopf algberas and transcendental numbers,' in Zeta Functions, Topology and Quantum Physics, Developments in Mathematics 14, T. Aoki et. al. (eds.), Springer, New York, 2005, pp. 197-220.
  54. T. Aoki and Y. Ohno, `Sum relations for multiple zeta values and connection formulas for the Gauss hypergeometric functions,' Publ. Res. Inst. Math. Sci. 41 (2005), 329-337; preprint NT/0307264.
  55. J. Choi and H. M. Srivastava, `Explicit evaluation of Euler and related sums,' Ramanujan Journal 10 (2005), 51-70.
  56. J-W. Son and D. S. Jang, `Explicit evaluations of special multiple zeta values ζ({4l+2}n) and ζ({4l}n),' Commun. Korean Math. Soc. 20 (2005), 247-257.
  57. R. Masri, `Multiple Dedekind zeta functions and evaluations of extended multiple zeta values,' J. Number Theory 115 (2005), 295-309.
  58. M. Kaneko, `Multiple zeta values,' Sugaku Expositions 18 (2005), 221-232. (Translation of Japanese original that appeared in Sugaku 54 (2002), 404-415.)
  59. S. Zlobin, `Generating functions for a multiple zeta function,' (Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2005, 55-59; English translation in Moscow Univ. Math. Bull. 60 (2005), 44-48.
  60. K. Ihara, M. Kaneko, and D. Zagier, `Derivation and double shuffle relations for multiple zeta values,' Compositio Math. 142 (2006), 307-338; preprint MPIM2004-100.
  61. J. Kajikawa, `Duality and double shuffle relations for multiple zeta values,' J. Number Theory 121 (2006), 1-6.
  62. M. E. Hoffman, `Quasi-symmetric functions, multiple zeta values, and rooted trees,' Oberwolfach Reports 3 (2006), 1259-1262; preprint QA/0609413.
  63. A. Besser and H. Furusho, `The double shuffle relations for p-adic multiple zeta values,' in Primes and Knots (Baltimore, 2003), T. Kohno and M. Morishita (eds.), Contemp. Math. 416, American Math. Soc., Providence, RI, 2006, pp. 9-29; preprint NT/0310177.
  64. H. Furusho, `Multiple zeta values and Grothendick-Teichmüller groups,' in Primes and Knots (Baltimore, 2003), T. Kohno and M. Morishita (eds.), Contemp. Math. 416, American Math. Soc., Providence, RI, 2006, pp. 49-82; preprint RIMS-1357.
  65. K. Matsumoto, `Analytic properties of multiple zeta-functions in several variables,' in Number Theory: Tradition and Modernization, Developments in Mathematics 15, W. Zhang and T. Tanigawa (eds.), Springer, New York, 2006, pp. 153-173.
  66. T. Terasoma, `Geometry of multiple zeta values,' in International Congress of Mathematicians (Madrid, 2006), Vol. II, European Mathematical Society, Zürich, 2006, pp. 627-635.
  67. B. Enriquez and F. Gavarini, `A formula for the logarithm of the KZ associator,' SIGMA 2 (2006), Paper 080.
  68. Y. Ohno and N. Wakabayashi, `Cyclic sum of multiple zeta values,' Acta Arithmetica 123 (2006), 289-295.
  69. R. Pemantle and C. Schneider, `When is 0.999... equal to 1?,' Amer. Math. Monthly 114 (2007), 344-350.
  70. H. Furusho, `p-Adic multiple zeta values II--Tannakian interpretations,' Amer. J. Math. 129 (2007), 1105-1144; preprint NT/0506117.
  71. Y. Ohno and J. Okuda, `On the sum formula for the q-analogue of non-strict multiple zeta values,' Proc. Amer. Math. Soc. 135 (2007), 3029-3037.
  72. M. Kaneko, `On an extension of the derivation relation for multiple zeta values,' in The Conference on L-Functions (Fukuoka, 2006), L. Weng and M. Kaneko (eds.), World Scientific, Hackensanck, NJ, 2007, pp. 89-94; preprint.
  73. D. M. Bradley, `On the sum formula for multiple q-zeta values,' Rocky Mountain J. Math. 37 (2007), 1427-1434; preprint QA/0411274.
  74. J. Zhao, `Multiple q-zeta functions and multiple q-polylogarithms,' Ramanujan J. 14 (2007), 189-221; preprint QA/0304448.
  75. J. Okuda and Y. Takeyama, `On relations for the multiple q-zeta values,' Ramanujan J. 14 (2007), 379-387; preprint QA/0402152.
  76. H. Furusho and A. Jafari, `Regularization and generalized double shuffle relations for p-adic multiple zeta values,' Compositio Math. 143 (2007), 1089-1107; preprint NT/0510681.
  77. T. Aoki, Y. Kombu, and Y. Ohno, `A generating function for sums of multiple zeta values and its applications,' Proc. Amer. Math. Soc. 136 (2008), 387-395.
  78. Z-h. Li, `Sum of multiple zeta values of fixed weight, depth, and i-height,' Math. Z. 258 (2008), 133-142.
  79. J. Okuda and K. Ueno, `New approach to Ohno relations for multiple zeta values,' preprint NT/0106148.
  80. S. Kitani, E. Sawada, and K. Ueno, `Finite automata and relations of multiple zeta values,' preprint NT/0403458.
  81. S. Oi, `Representation of the Gauss hypergeometric function by multiple polylogarithms and relations of multiple zeta values,' preprint NT/0405162.
  82. S. Zlobin, `A certain integral over a triangle,' preprint NT/0511239.
  83. S. Zlobin, `A note on arithmetic properties of multiple zeta values,' preprint NT/0501151.
  84. K. Ebrahimi-Fard and L. Guo, `Multiple zeta values and Rota-Baxter algebras,' preprint NT/0601558.
  85. F. C. S. Brown, `Multiple zeta values and periods of moduli spaces M0,n,' preprint AG/0606419.
  86. L. Guo and B. Zhang, `Renormalization of multiple zeta values,' preprint NT/0606076.
  87. J. Cresson, S. Fischler and T. Rivoal, `Phénomènes de symétrie dans des formes linéaires en polyzêtas,' preprint NT/0609744.
  88. S. Fischler, `Multiple series connected to Hoffman's conjecture on multiple zeta values,' preprint NT/0609799.
  89. I. Horozov, `Multiple zeta values, modular forms, and adeles,' preprint NT/0611849.
  90. J. Zhao, `Renormalization of multiple q-zeta values,' preprint NT/0612093.
  91. D. Manchon and S. Paycha, `Chen sums of symbols and renormalised multiple zeta functions,' preprint NT/0702135.
  92. G. Kawashima, `A class of relations among multiple zeta values,' preprint NT/0702824.
  93. J. Zhao, `An exotic shuffle relation of &zeta({2}m) and &zeta({3,1}n),' preprint 0707.3244[NT].
  94. L. Guo and B. Zhang, `Differential Birkhoff decomposition and the renormalization of multiple zeta values,' preprint 0710.0432[NT].
  95. S. Muneta, `On some explicit evaluations of multiple star-zeta values,' preprint 0710.3219[NT].
  96. T. Tanaka, `On extended derivation relations for multiple zeta values,' preprint 0710.4920[NT].
  97. S. Muneta, `Algebraic setup of non-strict multiple zeta values,' preprint 0711.0252[NT].
  98. M. Kaneko, `A note on poly-Bernoulli numbers and multiple zeta values,' preprint.
  99. S. Muneta, `A note on evaluations of multiple zeta values,' preprint 0802.4331[NT].
  100. Y. Ohno and W. Zudilin, `Zeta stars,' preprint MPIM2007-134.
  101. F. C. S. Brown, `The massless higher-loop two-point function,' preprint 0804.1660[AG].
  102. R. Lu, `The Γ^-genus and a regularization of an S1-equivariant Euler class,' preprint 0804.2714[math-ph].

D. ALTERNATING SERIES

  1. D. H. Bailey, J. M. Borwein, and R. Girgensohn, `Experimental evaluation of Euler sums,' Experiment. Math. 3 (1994), 17-30.
  2. D. Borwein, J. M. Borwein, and R. Girgensohn, `Explicit evaluation of Euler sums,' Proc. Edinburgh Math. Soc. 38 (1995), 277-294.
  3. V. Adamchik, `On Stirling numbers and Euler sums,' J. Comp. Appl. Math. 79 (1997), 119-130.
  4. D. J. Broadhurst, J. M. Borwein, and D. M. Bradley, `Evaluation of k-fold Euler/Zagier sums: a compendium of results for arbitrary k,' Electronic J. Combinatorics 4(2) (1997), R5.
  5. P. Flajolet and B. Salvy, `Euler sums and contour integral representations,' Experiment. Math. 7 (1998), 15-35.
  6. M. Bigotte, G. Jacob, N. E. Oussous, and M. Petitot, `Lyndon words and shuffle algebras for generating the coloured multiple zeta values relations tables,' Theor. Comput. Sci. 273 (2002), 271-283.
  7. D. Borwein, J. M. Borwein, and D. M. Bradley, `Parametric Euler sum identities,' J. Math. Anal. Appl. 316 (2006), 328-338.
  8. J. M. Borwein and D. M. Bradley, `Thirty-two Goldbach variations,' Intl. J. Number Theory 2 (2006), 65-103; preprint NT/0502034.
  9. D. J. Broadhurst, `On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory,' preprint hep-th9604128.
  10. D. J. Broadhurst, `Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams,' preprint hep-th9612012.
  11. M. Bigotte, G. Jacob, N. E. Oussous, and M. Petitot, `Tables des relations de la fonction zéta colorée,' LIFL Publication IT-322.
  12. M. N. Lalín, `On a certain combination of colored multizeta values,' preprint NT/0603442.
  13. J. Zhao, `Double shuffle relations of Euler sums,' preprint 0705.2267[NT].

E. MULTIPLE POLYLOGARITHMS/NESTED SUMS

  1. A. B. Goncharov, `Multiple polylogarithms, cyclotomy, and modular complexes,' Math. Res. Lett. 5 (1998), 497-516.
  2. D. J. Broadhurst, `Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity,' European Phys. J. C (Fields) 8 (1999), 311-333; preprint hep-th9803091.
  3. M. E. Hoffman, `Quasi-shuffle products,' J. Algebraic Combin. 11 (2000), 49-68; preprint.
  4. A. I. Davydychev and M. Yu. Kalmykov, `Some remarks on the ε-expansion of dimensionally regulated Feynman diagrams,' Nuclear Phys. B (Proc. Suppl.) 89 (2000), 283-288.
  5. A. I. Davydychev and M. Yu. Kalmykov, `New results for the ε-expansion of certain one-, two- and three-loop Feynman diagrams,' Nuclear Phys. B 605 (2001), 266-318.
  6. J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, `Special values of multidimensional polylogarithms,' Trans. Amer. Math. Soc. 353 (2001), 907-941.
  7. J. M. Borwein, D. J. Broadhurst, and J. Kamnitzer, `Central binomial sums, multiple Clausen values, and zeta values,' Experiment. Math. 10 (2001), 25-34.
  8. Hoang Ngoc Minh, G. Jacob, M. Petitot, and N. E. Oussous, `De l'algèbre des ζ de Riemann multivariées à l'algèbre des ζ de Hurwitz multivariées,' Sém. Lothar. Combin. 44 (2001), art. B44i.
  9. D. Bowman and D. M. Bradley, `Multiple polylogarithms: a brief survey,' in Conference on q-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000), Contemp. Math. 291, B. C. Berndt and K. Ono (eds.), Amer. Math. Soc., Providence, RI, 2001, pp. 71-92.
  10. G. Racinet, `Torseurs associés à certaines relations algébriques entre polyzêtas aux racines de l'unité,' C. R. Acad. Sci. Paris Ser. I 333 (2001), 5-10.
  11. G. Racinet, `Algèbre de Lie de valeuers formelles d'hyperlogarithmes aux racines de l'unité' C. R. Acad. Sci. Paris Ser. I 333 (2001), 11-16.
  12. S. Moch, P. Uwer, and S. Weinzierl, `Nested sums, expansion of transcendental functions and multiscale multiloop integrals,' J. Math. Phys. 43 (2002), 3363-3386; preprint hep-ph0110083.
  13. A. B. Goncharov, `Multiple ζ-values, Galois groups, and geometry of modular varieties,' in European Congress of Mathematics (Barcelona, 2000), Vol. I, Progr. Math. 201, Birkhäuser, Basel, 2001, pp. 361-392; preprint AG/0005069 .
  14. G. Racinet, `Doubles mélanges des polylogarithmes multiples aux racines de l'unité,' Publ. Math. IHES 95 (2002), 185-231; preprint QA/0202142; English translation (courtesy of D. Moskovich).
  15. M. Waldschmidt, `Multiple polylogarithms: an introduction,' in Number Theory and Discrete Mathematics (Chandigarh, 2000), Birkhäuser, Basel, 2000, pp. 1-12.
  16. M. Bigotte, G. Jacob, N. E. Oussous, and M. Petitot, `Coloured shuffle algebra and coloured polylogarithm functions,' LIFL Publication 2000-02.
  17. M. Lalín, `Some examples of Mahler measure as multiple polylogarithms,' J. Number Theory 103 (2003), 85-108.
  18. E. A. Ulanskii, `Identities for generalized polylogarithms,' (Russian), Mat. Zametki 73 (2003), 613-624; English translation in Math. Notes 73 (2003), 571-581.
  19. M. Kaneko and T. Arakawa, `On multiple L-values,' J. Math. Soc. Japan 56 (2004), 967-991.
  20. Hoang Ngoc Minh, `Shuffle algebra and differential Galois group of colored polylogarithms,' Nuclear Phys. B (Proc. Suppl.) 135 (2004), 220-224.
  21. J. Okuda, `Duality formulas of the special values of multiple polylogarithms,' Bull. London Math. Soc. 37 (2005), 230-242; preprint CA/0307137.
  22. J. Vollinga and S. Weinzierl, `Numerical evaluation of multiple polylogarithms,' Comput. Phys. Commun. 167 (2005), 177-194; preprint hep-ph0410259.
  23. Q. Wang, `Moduli spaces and multiple polylogarithm motives,' Adv. in Math. 206 (2006), 329-357; preprint AG/0610670.
  24. M. Yu. Kalmykov, B. F. L. Ward and S. Yost, `All order ε-expansion of Gauss hypergeometric functions with integer and half-integer values of parameters,' J. High Energy Phys. (2007), 02#040.
  25. M. Yu. Kalmykov, B. F. L. Ward and S. Yost, `Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter,' J. High Energy Phys. (2007), 10#048.
  26. K.-G. Schlesinger, `Some remarks on q-deformed multiple polylogarithms,' preprint QA/0111022.
  27. W. Zudilin, `One parameter models of Hopf algebras associated with multiple zeta values,' preprint.
  28. Yu. I. Manin, `Iterated integrals of modular forms and noncommutative modular symbols,' preprint AG/0502576.
  29. J. Sondow and S. Zlobin, `Integrals over polytopes, multiple zeta values and polylogarithms, and Euler's constant,' preprint 0705.0732[NT].
  30. J. Zhao, `Linear relations of special values of multiple polylogarithms at roots of unity,' preprint 0707.1459[NT].
  31. S. Zlobin, `Special values of generalized polylogarithms,' preprint 0712.1656[NT].

F. FINITE MULTIPLE HARMONIC SUMS

  1. J. Blümlein and S. Kurth, `Harmonic sums and Mellin transforms up to two-loop order,' Phys. Rev. D 60 (1999), art. 01418; preprint hep-ph9810241.
  2. J. A. M. Vermaseren, `Harmonic sums, Mellin transforms and integrals,' Int. J. Modern Phys. A 14 (1999), 2037-2076; preprint hep-ph9806280.
  3. S. Moch and J. A. M. Vermaseren, `Deep inelastic structure functions at two loops,' Nuclear Phys. B 573 (2000), 853-907; preprint hep-ph9912355.
  4. J. Blümlein, `Analytic continuation of Mellin transforms up to two-loop order,' Comput. Phys. Commun. 133 (2000), 76-104; preprint hep-ph0003100.
  5. J. Blümlein, `Algebraic relations between harmonic sums and associated quantities,' Comput. Phys. Commun. 159 (2004), 19-54; preprint hep-ph0311046.
  6. M. E. Hoffman, `The Hopf algebra structure of multiple harmonic sums,' Nuclear Phys. B (Proc. Suppl.) 135 (2004), 214-219; preprint QA/0406589.
  7. J. Blümlein and S. Moch, `Analytic continuation of the harmonic sums for the 3-loop anomalous dimensions,' Phys. Lett. B 614 (2005), 53-61; preprint hep-ph0503188.
  8. C. Sekine, `Partial sums of multiple zeta value series,' Tokyo J. Math. 29 (2006), 465-474.
  9. J. Zhao, `Bernoulli numbers, Wolstenholme's theorem, and p5 variations of Lucas' theorem,' J. Number Theory 123 (2007), 18-26.
  10. J. Zhao, `Multiple harmonic sums I: Generalizations of Wolstenholme's theorem,' preprint NT/0301252.
  11. J. Zhao, `Multiple harmonic sums II: Finiteness of p-divisible sets,' preprint NT/0303043.
  12. M. E. Hoffman, `Quasi-symmetric functions and mod p multiple harmonic sums,' preprint NT/0401319.

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