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
- P. H. Fuss (ed.), Correspondance
Mathématique et Physique de quelques célèbres
Géomètres (Tome 1), St. Petersburg, 1843.
- 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.
- N. Nielsen, Die Gammafunktion, Chelsea,
New York, 1965.
Reprint of Handbuch der Theorie der Gammafunktion (1906) and
Theorie der Integrallogarithmus und verwandter Transzendenten (1906).
- L. Tornheim, `Harmonic double series,' Amer. J. Math.
72(1950), 303-314.
- G. T. Williams, `A new method of evaluating
ζ(2n),' Amer. Math. Monthly 60 (1953), 19-25.
- T. M. Apostol and T. H. Vu, `Dirichlet series
related to the Riemann zeta function,' J. Number Theory 19 (1982),
85-102.
- M. V. Subbarao and R. Sitaramachandrarao,
`On some infinite series of L. J. Mordell and their analogues', Pacific J.
Math. 119 (1985), 245-255.
- R. E. Crandall and J. P. Buhler, `On the evaluation
of Euler sums,'
Experiment. Math. 3 (1994), 275-285.
- 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.
- J. G. Huard, K. S. Williams, and Zhang Nan-Yue,
`On Tornheim's double series,' Acta Arithmetica 75 (1996), 105-117.
- M-A. Coppo, `Sur les sommes d'Euler divergentes,'
Expositiones Mathematicae 18 (2000), 297-308.
- Chu Wenchang, `Symmetric functions and the Riemann
zeta series,' Indian J. Pure Appl. Math. 31 (2000), 1677-1689.
- K. Boyadzhiev, `Evaluation of Euler-Zagier sums,'
Internat. J. Math. Math. Sci. 27 (2001), 407-412.
- K. Boyadzhiev, `Consecutive evaluation of Euler sums,'
Internat. J. Math. Math. Sci. 29 (2002), 555-561.
- H. Tsumura, `On some combinatorial relations for
Tornheim's double series,' Acta Arithmetica 105 (2002), 239-252.
- 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.
- 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.
- H. Tsumura, `On alternating analogues of Tornheim's
double series,'
Proc. Amer. Math. Soc. 131 (2003), 3633-3641.
- H. Tsumura, `Evaluation formulas for Tornheim's
type of alternating double series,'
Math. Comp. 73 (2004), 251-258.
- H. Tsumura, `On evaluation formulas for double
L-values,' Bull. Austral. Math. Soc. 70 (2004), 213-221.
- D. Terhune, `Evaluation of double L-values,' J. Number
Theory 105 (2004), 275-301.
- R. Masri, `The Herglotz-Zagier function, double
zeta values, and values of L-series,' J. Number Theory 106 (2004),
219-237.
- 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.
- O. Espinosa and V. H. Moll, `The evaluation of
Tornheim double sums, Part I,' J. Number Theory 116 (2006), 200-229;
preprint CA/0505647.
- K-W. Chen and M. Eie, `Explicit evaluations of
extended Euler sums,' J. Number Theory 117 (2006), 31-52.
- 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.
- H. Tsumura, `On some functional relations between
Mordell-Tornheim double L-functions and Dirichlet L-functions,'
J. Number Theory 120 (2006), 161-178.
- 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.
- T. Nakamura, `A functional relation for the
Tornheim zeta function,' Acta Arithetica 125 (2006), 257-263.
- 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.
- J. M. Borwein, `Hilbert's inequality and Witten's
zeta-function," Amer. Math. Monthly 115 (2008), 125-137.
- 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.
- X. Zhou, T. Cai, and D. Bradley, `Signed q-analogs
of Tornheim's double series,'
Proc. Amer. Math. Soc. 136 (2008), 2689-2698.
- T. Machide, `Generators for vector spaces
consisting of double zeta values with even weight,'
preprint 0802.1565[NT].
B. TRIPLE HARMONIC SERIES
- R. Sitaramachandrarao and M. V. Subbarao,
`Transformation formulae for multiple series,' Pacific J. Math. 113
(1984), 471-479.
- C. Markett, `Triple sums and the Riemann zeta
function,' J. Number Theory 48 (1994), 113-132.
- 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.
- M. E. Hoffman and C. Moen, `Sums of triple harmonic
series,' J. Number Theory 60 (1996), 329-331.
- A. Panholzer and H. Prodinger, `Computer-free evaluation
of an infinite double sum via Euler sums,'
Sém. Lothar. Combin. 55 (2005), art. B55a.
- 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
- M. E. Hoffman, `Multiple harmonic series,' Pacific J.
Math. 152 (1992), 275-290.
- 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.
- 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.
- T. Q. T. Le and J. Murakami, `Kontsevich's integral
for the Kauffman polynomial,' Nagoya Math. J. 142 (1996), 39-65.
- 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.
- 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.
- M. E. Hoffman, `The algebra of multiple harmonic
series,' J. Algebra 194 (1997), 477-495.
- R. E. Crandall, `Fast evaluation of multiple zeta
sums,'
Math. Comp. 67 (1998), 1163-1172.
- J. M. Borwein, D. M. Bradley, D. J. Broadhurst,
and P. Lisonek, `Combinatorial aspects of multiple zeta values,'
Electronic J. Combinatorics 5 (1998), R38.
- 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.
- Y. Ohno, `A generalization of the duality and sum
formulas on the multiple zeta values,' J. Number Theory 74 (1999),
39-43.
- T. Arakawa and M. Kaneko, `Multiple zeta values,
poly-Bernoulli numbers,
and related zeta functions,' Nagoya Math. J. 153 (1999), 189-209.
- T. Takamuki, `The Kontsevich invariant and relations
of multiple zeta values,' Kobe J. Math. 16 (1999), 27-43.
- 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.
- J. Zhao, `Analytic continuation of multiple zeta
functions,'
Proc. Amer. Math. Soc. 128 (2000), 1275-1283.
- Hoang Ngoc Minh and M. Petitot, `Lyndon words,
polylogarithms, and the Riemann ζ function,' Discrete Math. 217
(2000), 273-292.
- M. Waldschmidt, `Valeurs zêta multiples. Une
introduction,' J. Théor. Nombres Bordeaux 12 (2000), 581-595.
- M. Kontsevich and D. Zagier, `Periods,' in
Mathematics Unlimited--2001 and Beyond, Springer, Berlin, 2001,
pp. 771-808.
- 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.
- K. Ihara and T. Takamuki, `The quantum g2
invariant and relations of multiple zeta values,' J. Knot Theory
Ramifications 10 (2001), 983-997.
- S. Akiyama and Y. Tanigawa,
`Multiple zeta values at non-positive integers,'
Ramanujan J. 5 (2001), 327-351.
- Y. Ohno and D. Zagier, `Multiple zeta values of
fixed weight, depth, and height,' Indag. Math. (N. S.) 12 (2001),
483-487.
- 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.
- M. E. Hoffman, `Periods of mirrors and multiple zeta
values,'
Proc. Amer. Math. Soc. 130 (2002), 971-974.
- T. Terasoma, `Selberg integrals and multiple zeta
values,' Compositio Math. 133 (2002), 1-24.
- P. Cartier, `Fonctions polylogarithmes, nombres
polyzêtas et groupes pro-unipotents,' Astérisque 282
(2002), 137-173 (Sém. Bourbaki no. 885).
- 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.
- T. Terasoma, `Mixed Tate motives and multiple zeta
values,' Invent. Math. 149 (2002), 339-369;
preprint AG/0104231.
- S. Fischler, `Formes linéaires en
polyzêtas et intégrales multiples,' C. R. Acad. Sci. Paris,
Ser. I 335 (2002), 1-4.
- 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.
- 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.
- M. E. Hoffman and Y. Ohno, `Relations of multiple
zeta values and their algebraic expression,' J. Algebra 262 (2003),
332-347;
preprint QA/0010140.
- 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.
- H. Ishikawa and K. Matsumoto, `On the estimation
of the order of
Euler-Zagier multiple zeta-functions,' Illinois J. Math. 47 (2003),
1151-1166.
- 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.
- D. Bowman and D. M. Bradley, `Resolution of
some open problems concerning multiple zeta evaluations of arbitrary depth,'
Compositio Math. 139 (2003), 85-100.
- H. Furusho, `The multiple zeta value algebra and
the stable derivation algebra,' Publ. Res. Inst. Math. Sci. 39 (2003),
695-720;
preprint NT/0011261.
- 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.
- H. Tsumura, `Combinatorial relations for Euler-Zagier
sums,' Acta Arithmetica 111 (2004), 27-42.
- H. Tsumura, `Multiple harmonic series related to
multiple Euler numbers,' J. Number Theory 106 (2004), 155-168.
- 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.
- S. Ünver, `p-Adic multi-zeta values,'
J. Number Theory 108 (2004), 111-156.
- 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.
- J. Écalle, `Multizetas, perinomal numbers,
arithmetical dimorphy,
and ARI/GARI,' Ann. Fac. Sci. Toulouse 13 (2004), 683-708.
- A. B. Goncharov and Yu. I. Manin, `Multiple
ζ-motives and moduli spaces M0,n,' Compositio Math.
140 (2004), 1-14;
preprint AG/0204102.
- D. M. Bradley, `Multiple q-zeta values,'
J. Algebra 283 (2005), 752-798; preprint QA/0402093.
- H. Tsumura, `On Mordell-Tornheim zeta values,'
Proc. Amer. Math. Soc. 133 (2005), 2387-2393.
- T. Nakamura, `Bernoulli numbers and multiple zeta
values,' Proc. Japan Acad. Ser. A 81 (2005), 21-22.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- J. Choi and H. M. Srivastava, `Explicit evaluation
of Euler and related sums,' Ramanujan Journal 10 (2005), 51-70.
- 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.
- R. Masri, `Multiple Dedekind zeta functions
and evaluations of
extended multiple zeta values,' J. Number Theory 115 (2005), 295-309.
- M. Kaneko, `Multiple zeta values,' Sugaku Expositions
18 (2005), 221-232. (Translation of Japanese original that appeared
in Sugaku 54 (2002), 404-415.)
- 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.
- 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.
- J. Kajikawa, `Duality and double shuffle relations
for multiple zeta values,' J. Number Theory 121 (2006), 1-6.
- M. E. Hoffman, `Quasi-symmetric functions,
multiple zeta values, and rooted trees,' Oberwolfach Reports 3
(2006), 1259-1262;
preprint
QA/0609413.
- 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.
- 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.
- 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.
- T. Terasoma, `Geometry of multiple zeta values,' in
International Congress of Mathematicians (Madrid, 2006), Vol. II,
European Mathematical Society, Zürich, 2006, pp. 627-635.
- B. Enriquez and F. Gavarini, `A formula for the
logarithm of the
KZ associator,'
SIGMA 2 (2006), Paper 080.
- Y. Ohno and N. Wakabayashi, `Cyclic sum of multiple
zeta values,' Acta Arithmetica 123 (2006), 289-295.
- R. Pemantle and C. Schneider, `When is 0.999... equal
to 1?,' Amer. Math. Monthly 114 (2007), 344-350.
- H. Furusho, `p-Adic multiple zeta values
II--Tannakian interpretations,' Amer. J. Math. 129 (2007), 1105-1144;
preprint NT/0506117.
- 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.
- 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.
- D. M. Bradley, `On the sum formula for multiple
q-zeta values,' Rocky Mountain J. Math. 37 (2007), 1427-1434;
preprint
QA/0411274.
- J. Zhao, `Multiple q-zeta functions and multiple
q-polylogarithms,' Ramanujan J. 14 (2007), 189-221;
preprint
QA/0304448.
- J. Okuda and Y. Takeyama, `On relations for the
multiple q-zeta values,' Ramanujan J. 14 (2007), 379-387;
preprint
QA/0402152.
- 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.
- 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.
- Z-h. Li, `Sum of multiple zeta values
of fixed weight, depth, and i-height,' Math. Z. 258 (2008),
133-142.
- J. Okuda and K. Ueno, `New approach to Ohno
relations for multiple zeta values,'
preprint NT/0106148.
- S. Kitani, E. Sawada, and K. Ueno, `Finite automata
and relations of multiple zeta values,'
preprint
NT/0403458.
- S. Oi, `Representation of the Gauss
hypergeometric function by multiple polylogarithms and relations of
multiple zeta values,'
preprint
NT/0405162.
- S. Zlobin, `A certain integral over a triangle,'
preprint
NT/0511239.
- S. Zlobin, `A note on arithmetic properties of
multiple zeta values,'
preprint
NT/0501151.
- K. Ebrahimi-Fard and L. Guo, `Multiple zeta values
and Rota-Baxter algebras,'
preprint
NT/0601558.
- F. C. S. Brown, `Multiple zeta values and periods
of moduli spaces M0,n,'
preprint
AG/0606419.
- L. Guo and B. Zhang, `Renormalization of multiple
zeta values,'
preprint
NT/0606076.
- J. Cresson, S. Fischler and T. Rivoal,
`Phénomènes de symétrie dans des formes
linéaires en polyzêtas,'
preprint
NT/0609744.
- S. Fischler, `Multiple series connected to
Hoffman's conjecture on multiple zeta values,'
preprint
NT/0609799.
- I. Horozov, `Multiple zeta values, modular forms,
and adeles,'
preprint
NT/0611849.
- J. Zhao, `Renormalization of multiple q-zeta values,'
preprint
NT/0612093.
- D. Manchon and S. Paycha, `Chen sums of symbols
and renormalised multiple zeta functions,'
preprint
NT/0702135.
- G. Kawashima, `A class of relations among multiple
zeta values,'
preprint
NT/0702824.
- J. Zhao, `An exotic shuffle relation of
&zeta({2}m) and &zeta({3,1}n),'
preprint
0707.3244[NT].
- L. Guo and B. Zhang, `Differential Birkhoff
decomposition and the renormalization of multiple zeta values,'
preprint
0710.0432[NT].
- S. Muneta, `On some explicit evaluations of
multiple star-zeta values,'
preprint
0710.3219[NT].
- T. Tanaka, `On extended derivation relations for
multiple zeta values,'
preprint
0710.4920[NT].
- S. Muneta, `Algebraic setup of non-strict multiple
zeta values,'
preprint
0711.0252[NT].
- M. Kaneko, `A note on poly-Bernoulli numbers and
multiple zeta values,'
preprint.
- S. Muneta, `A note on evaluations of multiple
zeta values,'
preprint
0802.4331[NT].
- Y. Ohno and W. Zudilin, `Zeta stars,'
preprint MPIM2007-134.
- F. C. S. Brown, `The massless higher-loop
two-point function,'
preprint
0804.1660[AG].
- R. Lu, `The Γ^-genus and a regularization
of an S1-equivariant Euler class,'
preprint
0804.2714[math-ph].
D. ALTERNATING SERIES
- D. H. Bailey, J. M. Borwein, and R. Girgensohn,
`Experimental evaluation of Euler sums,'
Experiment. Math. 3 (1994), 17-30.
- D. Borwein, J. M. Borwein, and R. Girgensohn,
`Explicit evaluation of Euler sums,' Proc. Edinburgh Math. Soc. 38
(1995), 277-294.
- V. Adamchik, `On Stirling numbers and Euler sums,'
J. Comp. Appl. Math. 79 (1997), 119-130.
- 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.
- P. Flajolet and B. Salvy, `Euler sums and contour
integral representations,'
Experiment. Math. 7 (1998), 15-35.
- 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.
- D. Borwein, J. M. Borwein, and D. M. Bradley,
`Parametric Euler sum identities,' J. Math. Anal. Appl. 316 (2006),
328-338.
- J. M. Borwein and D. M. Bradley, `Thirty-two
Goldbach variations,' Intl. J. Number Theory 2 (2006), 65-103;
preprint NT/0502034.
- D. J. Broadhurst, `On the enumeration of
irreducible k-fold Euler sums and their roles in knot theory and
field theory,'
preprint hep-th9604128.
- D. J. Broadhurst, `Conjectured enumeration of
irreducible multiple zeta values, from knots and Feynman diagrams,'
preprint hep-th9612012.
- M. Bigotte, G. Jacob, N. E. Oussous, and M. Petitot,
`Tables des relations de la fonction zéta colorée,'
LIFL Publication IT-322.
- M. N. Lalín, `On a certain combination of
colored multizeta values,'
preprint
NT/0603442.
- J. Zhao, `Double shuffle relations of Euler sums,'
preprint 0705.2267[NT].
E. MULTIPLE POLYLOGARITHMS/NESTED SUMS
- A. B. Goncharov, `Multiple polylogarithms, cyclotomy,
and modular complexes,' Math. Res. Lett. 5 (1998), 497-516.
- 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.
- M. E. Hoffman, `Quasi-shuffle products,' J. Algebraic
Combin. 11 (2000), 49-68; preprint.
- 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.
- 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.
- 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.
- J. M. Borwein, D. J. Broadhurst, and J. Kamnitzer,
`Central binomial sums, multiple Clausen values, and zeta values,'
Experiment. Math. 10 (2001), 25-34.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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
.
- 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).
- M. Waldschmidt, `Multiple polylogarithms: an
introduction,' in
Number Theory and Discrete Mathematics (Chandigarh, 2000),
Birkhäuser, Basel, 2000, pp. 1-12.
- M. Bigotte, G. Jacob, N. E. Oussous, and M. Petitot,
`Coloured shuffle algebra and coloured polylogarithm functions,'
LIFL Publication
2000-02.
- M. Lalín, `Some examples of Mahler measure
as multiple polylogarithms,' J. Number Theory 103 (2003), 85-108.
- E. A. Ulanskii, `Identities for generalized
polylogarithms,' (Russian),
Mat. Zametki 73 (2003), 613-624; English translation in Math.
Notes 73 (2003), 571-581.
- M. Kaneko and T. Arakawa, `On multiple L-values,'
J. Math. Soc. Japan 56 (2004), 967-991.
- Hoang Ngoc Minh, `Shuffle algebra and differential
Galois group of colored polylogarithms,' Nuclear Phys. B (Proc. Suppl.)
135 (2004), 220-224.
- J. Okuda, `Duality formulas of the special values
of multiple polylogarithms,' Bull. London Math. Soc. 37 (2005), 230-242;
preprint CA/0307137.
- J. Vollinga and S. Weinzierl, `Numerical evaluation
of multiple polylogarithms,' Comput. Phys. Commun. 167 (2005), 177-194;
preprint hep-ph0410259.
- Q. Wang, `Moduli spaces and multiple polylogarithm
motives,' Adv. in Math. 206 (2006), 329-357;
preprint
AG/0610670.
- 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.
- 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.
- K.-G. Schlesinger, `Some remarks on q-deformed
multiple polylogarithms,'
preprint
QA/0111022.
- W. Zudilin, `One parameter models of Hopf
algebras associated with multiple zeta values,'
preprint.
- Yu. I. Manin, `Iterated integrals of modular forms
and noncommutative modular symbols,'
preprint AG/0502576.
- J. Sondow and S. Zlobin, `Integrals over polytopes,
multiple zeta values and polylogarithms, and Euler's constant,'
preprint 0705.0732[NT].
- J. Zhao, `Linear relations of special values
of multiple polylogarithms at roots of unity,'
preprint 0707.1459[NT].
- S. Zlobin, `Special values of generalized
polylogarithms,'
preprint 0712.1656[NT].
F. FINITE MULTIPLE HARMONIC SUMS
- 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.
- J. A. M. Vermaseren, `Harmonic sums, Mellin transforms
and integrals,' Int. J. Modern Phys. A 14 (1999), 2037-2076;
preprint hep-ph9806280.
- S. Moch and J. A. M. Vermaseren, `Deep inelastic
structure functions at two loops,' Nuclear Phys. B 573 (2000), 853-907;
preprint hep-ph9912355.
- J. Blümlein, `Analytic continuation of Mellin
transforms up to two-loop order,' Comput. Phys. Commun. 133 (2000),
76-104;
preprint hep-ph0003100.
- J. Blümlein, `Algebraic relations between
harmonic sums and associated quantities,' Comput. Phys. Commun. 159
(2004), 19-54;
preprint hep-ph0311046.
- M. E. Hoffman, `The Hopf algebra structure of
multiple harmonic sums,' Nuclear Phys. B (Proc. Suppl.) 135 (2004),
214-219;
preprint QA/0406589.
- 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.
- C. Sekine, `Partial sums of multiple zeta value
series,' Tokyo J. Math. 29 (2006), 465-474.
- J. Zhao, `Bernoulli numbers, Wolstenholme's
theorem, and p5 variations of Lucas' theorem,'
J. Number Theory 123 (2007), 18-26.
- J. Zhao, `Multiple harmonic sums I: Generalizations of
Wolstenholme's theorem,'
preprint NT/0301252.
- J. Zhao, `Multiple harmonic sums II: Finiteness of
p-divisible sets,'
preprint NT/0303043.
- M. E. Hoffman, `Quasi-symmetric functions and
mod p multiple harmonic sums,'
preprint NT/0401319.
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