Geodesic
Special values of generalized polylogarithms
Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 6, pp. 63-89.

Voir la notice de l'article provenant de la source Math-Net.Ru

The values of generalized polylogarithms at various points and their relationships are examined. A detailed investigation is given of the polylogarithms of small weight at the points $1/2$ and $-1$. A conjecture about the structure of the linear space generated by values of generalized polylogarithms is put forward. 
@article{FPM_2010_16_6_a5,
     author = {S. A. Zlobin},
     title = {Special values of generalized polylogarithms},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {63--89},
     publisher = {mathdoc},
     volume = {16},
     number = {6},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2010_16_6_a5/}
}
TY  - JOUR
AU  - S. A. Zlobin
TI  - Special values of generalized polylogarithms
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2010
SP  - 63
EP  - 89
VL  - 16
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2010_16_6_a5/
LA  - ru
ID  - FPM_2010_16_6_a5
ER  - 
%0 Journal Article
%A S. A. Zlobin
%T Special values of generalized polylogarithms
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2010
%P 63-89
%V 16
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2010_16_6_a5/
%G ru
%F FPM_2010_16_6_a5
S. A. Zlobin. Special values of generalized polylogarithms. Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 6, pp. 63-89. http://geodesic.mathdoc.fr/item/FPM_2010_16_6_a5/

[1] Beitman G., Erdeii A., Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Fizmatgiz, M., 1965

[2] Zlobin S. A., “Razlozheniya kratnykh integralov v lineinye formy”, Mat. zametki, 77:5 (2005), 683–706 | DOI | MR | Zbl

[3] Zlobin S. A., Sondou Dzh., “Integraly po mnogogrannikam, kratnye dzeta znacheniya i polilogarifmy, i konstanta Eilera”, Mat. zametki, 84:4 (2008), 609–626 | DOI | MR | Zbl

[4] Zudilin V. V., “Algebraicheskie sootnosheniya dlya kratnykh dzeta-znachenii”, Uspekhi mat. nauk, 58:1 (2003), 3–32 | DOI | MR | Zbl

[5] Sorokin V. N., “O lineinoi nezavisimosti znachenii obobschënnykh polilogarifmov”, Mat. sbornik, 192:8 (2001), 139–154 | DOI | MR | Zbl

[6] Ulanskii E. A., “Tozhdestva dlya obobschënnykh polilogarifmov”, Mat. zametki, 73:4 (2003), 613–624 | DOI | MR | Zbl

[7] Abramowitz M., Stegun I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972

[8] Berndt B., Ramanujan's Notebooks, Part 1, Springer, Berlin, 1989 | MR | Zbl

[9] Bigotte M., Jacob G., Hoang Ngoc Minh, Oussous N. E., Petitot M., Algèbre des nombres d'Euler–Zagier, calculs effectifs et conjectures, preprint, Lille, 1999 http://www2.lifl.fr/~ejc2002/local/polylogs.ps

[10] Borwein J. M., Bradley D. M., Broadhurst D. J., “Evaluations of $k$-fold Euler/Zagier sums: A compendium of results for arbitrary $k$”, Electron. J. Combin., 4:2 (1997), Research Paper 5 | MR | Zbl

[11] Borwein J. M., Bradley D. M., Broadhurst D. J., Lisoněk P., “Special values of multiple polylogarithms”, Trans. Am. Math. Soc., 353:3 (2001), 907–941 | DOI | MR | Zbl

[12] Bowman D., Bradley D. M., “Resolution of some open problems concerning multiple zeta evaluations of arbitrary depth”, Compositio Math., 139:1 (2003), 85–100 | DOI | MR | Zbl

[13] Broadhurst D. J., Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams, 1996, arXiv: hep-th/9612012

[14] Broadhurst D. J., On the enumeration of irreducible $k$-fold Euler sums and their roles in knot theory and field theory, 1996, arXiv: hep-th/9604128

[15] Deligne P., Goncharov A. B., “Groupes fondamentaux motiviques de Tate mixte”, Ann. Sci. École Norm. Sup., 38:1 (2005), 1–56 | MR | Zbl

[16] Ferguson H. R. P., Bailey D. H., Arno S., “Analysis of PSLQ, an integer relation finding algorithm”, Math. Comp., 68 (1999), 351–369 | DOI | MR | Zbl

[17] Flajolet P., Salvy B., “Euler sums and contour integral representations”, Experiment. Math., 7:1 (1998), 15–35 | DOI | MR | Zbl

[18] Hoffman M. E., “The algebra of multiple harmonic series”, J. Algebra, 194:2 (1997), 477–495 | DOI | MR | Zbl

[19] Lewin L., Polylogarithms and Associated Functions, North-Holland, New York, 1981 | MR | Zbl

[20] http://mathworld.wolfram.com/BernoulliNumber.html

[21] Hoang Ngoc Minh, Petitot M., “Lyndon words, polylogarithms and the Riemann $\zeta$ function”, Discrete Math., 217:1–3 (2000), 273–292 | MR | Zbl

[22] Hoang Ngoc Minh, Petitot M., van der Hoeven J., L'algèbre des polylogarithmes par les séries génératrices, 1999 http://www.lifl.fr/~petitot/

[23] Nielsen N., Handbuch der Theorie der Gammafunktion, Chelsea, New York, 1965

[24] Terasoma T., “Mixed Tate motives and multiple zeta values”, Invent. Math., 149 (2002), 339–369, arXiv: math.AG/0104231 | DOI | MR | Zbl

[25] Vermaseren J. A. M., “Harmonic sums, Mellin transforms and Integrals”, Internat. J. Modern Phys. A, 14 (1999), 2037–2076, arXiv: hep-ph/9806280 | DOI | MR | Zbl

[26] Waldschmidt M., “Multiple polylogarithms: an introduction”, Number Theory and Discrete Mathematics, Proc. of the Int. Conf. in Honour of Srinivasa Ramanujan (Chandigarh, 2000), eds. A. K. Agarwal et al., Birkhäuser, Basel, 2002, 1–12 | DOI | MR | Zbl

[27] Zagier D., “Values of zeta functions and their applications”, First European Congress of Mathematics (Paris, 1992), v. II, Progress Math., 120, eds. A. Joseph et al., Birkhäuser, Boston, 1994, 497–512 | MR | Zbl