×

Shuffle products for multiple zeta values and partial fraction decompositions of zeta-functions of root systems. (English) Zbl 1231.11104

The shuffle product is defined on a certain set of polynomials over \(\mathbb Q\) in two non-commutative indeterminates making this set a commutative algebra and providing an algebraic interpretation to some relations for multiple zeta values (MZVs); see M. E. Hoffman and Y. Ohno [J. Algebra 262, No. 2, 332–347 (2003; Zbl 1139.11322)].
The authors give a new interpretation of the shuffle product expressing it in terms of the zeta values of root systems. This leads to a new proof of the double shuffle relations [K. Ihara, M. Kaneko and D. Zagier, Compos. Math. 142, No. 2, 307–338 (2006; Zbl 1186.11053)] avoiding the use of Drinfeld’s integral expressions for MZVs. Some functional relations for zeta functions of root systems (involving the shuffle products) are also given.

MSC:

11M32 Multiple Dirichlet series and zeta functions and multizeta values
17B22 Root systems
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Akiyama S., Egami S., Tanigawa Y.: Analytic continuation of multiple zeta functions and their values at non-positive integers. Acta Arith. 98, 107–116 (2001) · Zbl 0972.11085 · doi:10.4064/aa98-2-1
[2] Arakawa T., Kaneko M.: Multiple zeta values, poly-Bernoulli numbers and related zeta functions. Nagoya Math. J. 153, 189–209 (1999) · Zbl 0932.11055
[3] Arakawa T., Kaneko M.: On multiple L-values. J. Math. Soc. Jpn. 56, 967–991 (2004) · Zbl 1065.11068 · doi:10.2969/jmsj/1190905444
[4] Borwein J.M., Bradley D.M., Broadhurst D.J., Lisonek P.: Combinatorial aspects of multiple zeta values. Electron. J. Comb. 5, R38 (1998) · Zbl 0904.05012
[5] Borwein J.M., Bradley D.M., Broadhurst D.J., Lisonek P.: Special values of multidimensional polylogarithms. Trans. Am. Math. Soc. 353, 907–941 (2001) · Zbl 1002.11093 · doi:10.1090/S0002-9947-00-02616-7
[6] Bowman, D., Bradley, D.M.: Multiple polylogarithms: a brief survey. In: Berndt, B.C., Ono, K. (eds.) Conference on q-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000). Contemp. Math., vol. 291, pp. 71–92. Amer. Math. Soc., Providence (2001) · Zbl 0998.33013
[7] Bowman D., Bradley D.M.: The algebra and combinatorics of shuffles and multiple zeta values. J. Comb. Theory Ser. A 97, 43–61 (2002) · Zbl 1021.11026 · doi:10.1006/jcta.2001.3194
[8] Bowman, D., Bradley, D.M.: Resolution of some open problems concerning multiple zeta evaluations of arbitrary depth. Compos. Math. 139, 85–100 (2003) · Zbl 1035.11037 · doi:10.1023/B:COMP.0000005036.52387.da
[9] Bradley, D.M.: Partition identities for the multiple zeta function. In: Aoki, T. et al. (eds.) Zeta Functions, Topology and Quantum Physics, Developments in Mathematics, vol. 14, pp. 19–29. Springer, New York (2005) · Zbl 1170.11323
[10] Drinfel’d, V.G.: On quasitriangular quasi-Hopf algebras and a group closely related with Gal $${(\(\backslash\)overline{{\(\backslash\)bf Q}}/{\(\backslash\)bf Q})}$$ . Algebra i Analiz 2(4), 149–181 (1990) (in Russian); Translation in Leningr. Math. J. 2(4), 829–860 (1991)
[11] Essouabri D.: Singularités des séries de Dirichlet associées à des polynômes de plesieurs variables et applications en théorie analytique des nombres. Annales de L’Institut Fourier (Grenoble) 47, 429–483 (1997) · Zbl 0882.11051 · doi:10.5802/aif.1570
[12] Goncharov, A.B.: Periods and mixed motives. Preprint. arXiv:math/0202154 (2002)
[13] Huard J.G., Williams K.S., Zhang N.-Y.: On Tornheim’s double series. Acta Arith. 75, 105–117 (1996) · Zbl 0858.40008
[14] Hoffman M.E.: Multiple harmonic series. Pac. J. Math. 152, 275–290 (1992) · Zbl 0763.11037 · doi:10.2140/pjm.1992.152.275
[15] Hoffman M.E.: The algebra of multiple harmonic series. J. Algebra 194, 477–495 (1997) · Zbl 0881.11067 · doi:10.1006/jabr.1997.7127
[16] Hoffman, M.E.: Algebraic aspects of multiple zeta values. In: Aoki, T. et al. (eds.) Zeta Functions, Topology and Quantum Physics, Developments in Mathematics, vol. 14, pp. 51–74. Springer, New York, (2005) · Zbl 1170.11324
[17] Hoffman M.E., Ohno Y.: Relations of multiple zeta values and their algebraic expression. J. Algebra 262, 332–347 (2003) · Zbl 1139.11322 · doi:10.1016/S0021-8693(03)00016-4
[18] Ihara K., Kaneko M., Zagier D.: Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142, 307–338 (2006) · Zbl 1186.11053 · doi:10.1112/S0010437X0500182X
[19] Kaneko M.: Multiple zeta values. Sugaku Expo. 18, 221–232 (2005) · Zbl 1218.11080
[20] Komori Y., Matsumoto K., Tsumura H.: Zeta-functions of root systems. In: Weng, L., Kaneko, M. (eds) The Conference on L-functions (Fukuoka 2006), pp. 115–140. World Scientific Publishers, Hackensack (2007) · Zbl 1183.11055
[21] Komori Y., Matsumoto K., Tsumura H.: Zeta and L-functions and Bernoulli polynomials of root systems. Proc. Jpn. Acad. Ser. A 84, 57–62 (2008) · Zbl 1147.11053 · doi:10.3792/pjaa.84.57
[22] Komori Y., Matsumoto K., Tsumura H.: On multiple Bernoulli polynomials and multiple L-functions of root systems. Proc. Lond. Math. Soc. 100, 303–347 (2010) · Zbl 1217.11078 · doi:10.1112/plms/pdp025
[23] Komori Y., Matsumoto K., Tsumura H.: On Witten multiple zeta-functions associated with semisimple Lie algebras II. J. Math. Soc. Jpn. 62, 355–394 (2010) · Zbl 1210.11099 · doi:10.2969/jmsj/06220355
[24] Komori, Y., Matsumoto, K., Tsumura, H.: On Witten multiple zeta-functions associated with semisimple Lie algebras III. Preprint. arXiv:math/0907.0955 · Zbl 1270.11090
[25] Markett C.: Triple sums and the Riemann zeta function. J. Number Theory 48, 113–132 (1994) · Zbl 0810.11047 · doi:10.1006/jnth.1994.1058
[26] Matsumoto K.: On the analytic continuation of various multiple zeta-functions. In: Bennett, M.A et al. (eds) Number Theory for the Millennium II, pp. 417–440. AK Peters, Massachusetts (2002) · Zbl 1031.11051
[27] Matsumoto K.: Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series. Nagoya Math. J. 172, 59–102 (2003) · Zbl 1060.11053
[28] Matsumoto, K.: On Mordell-Tornheim and other multiple zeta-functions. In: Heath-Brown, D.R., Moroz, B.Z. (eds.) Proceedings of the Session in Analytic Number Theory and Diophantine Equations (Bonn, 2002), no. 25, 17 pp. Bonner Math. Schriften 360 (2003) · Zbl 1056.11049
[29] Matsumoto, K.: Analytic properties of multiple zeta-functions in several variables. In: Zhang, W., Tanigawa, Y. (eds.) Number Theory: Tradition and Modernization, Proceedings of the 3rd China-Japan Seminar, pp. 153-173. Springer, Berlin (2006) · Zbl 1197.11120
[30] Matsumoto K., Tsumura H.: On Witten multiple zeta-functions associated with semisimple Lie algebras I. Annales de L’Institut Fourier (Grenoble) 56, 1457–1504 (2006) · Zbl 1168.11036 · doi:10.5802/aif.2218
[31] Matsumoto K., Tsumura H.: A new method of producing functional relations among multiple zeta-functions. Q. J. Math. 59, 55–83 (2008) · Zbl 1151.11045 · doi:10.1093/qmath/ham025
[32] Mordell L.J.: On the evaluation of some multiple series. J. Lond. Math. Soc. 33, 368–371 (1958) · Zbl 0081.27501 · doi:10.1112/jlms/s1-33.3.368
[33] Ohno Y.: A generalization of the duality and sum formulas on the multiple zeta values. J. Number Theory 74, 39–43 (1999) · Zbl 0920.11063 · doi:10.1006/jnth.1998.2314
[34] Ohno Y., Zagier D.: Multiple zeta values of fixed weight, depth, and height. Indag. Math. (N. S.) 12, 483–487 (2001) · Zbl 1031.11053 · doi:10.1016/S0019-3577(01)80037-9
[35] Reutenauer C.: The shuffle algebra on the factors of a word is free. J. Comb. Theory Ser. A 38, 48–57 (1985) · Zbl 0565.68074 · doi:10.1016/0097-3165(85)90020-2
[36] Reutenauer C.: Free Lie Algebra. Oxford Science Publications, London (1993) · Zbl 0798.17001
[37] Terasoma T.: Mixed Tate motives and multiple zeta values. Invent. Math. 149, 339–369 (2002) · Zbl 1042.11043 · doi:10.1007/s002220200218
[38] Tornheim L.: Harmonic double series. Am. J. Math. 72, 303–314 (1950) · Zbl 0036.17203 · doi:10.2307/2372034
[39] Tsumura H.: On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function. Math. Proc. Camb. Philos. Soc. 142, 395–405 (2007) · Zbl 1149.11044 · doi:10.1017/S0305004107000059
[40] Witten E.: On quantum gauge theories in two dimensions. Commun. Math. Phys. 141, 153–209 (1991) · Zbl 0762.53063 · doi:10.1007/BF02100009
[41] Zagier, D.: Values of zeta functions and their applications. In: First European Congress of Mathematics, vol. II (Paris, 1992), pp. 497–512. Progress in Mathematics 120. Birkhäuser, Basel (1994) · Zbl 0822.11001
[42] Zhao J.: Analytic continuation of multiple zeta functions. Proc. Am. Math. Soc. 128, 1275–1283 (2000) · Zbl 0949.11042 · doi:10.1090/S0002-9939-99-05398-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.