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The algebra and combinatorics of shuffles and multiple zeta values. (English) Zbl 1021.11026
The authors consider multiple zeta values \[ \zeta (s_1,\ldots ,s_k)=\sum_{n_1>n_2>\ldots >n_k>0} \prod_{j=1}^kn_j^{-s_j} \] where \(s_1,\ldots ,s_k\) are positive integers, \(s_1>1\).
In particular, given a vector \(\vec s=(m_0,m_1,\ldots ,m_{2n})\) of non-negative integers, they study \[ Z(\vec s)=\zeta \left( \{2\}^{m_0},3,\{2\}^{m_1},1,\{2\}^{m_2},3,\{2\}^{m_3},1,\ldots ,3,\{2\}^{m_{2n-1}},1,\{2\}^{m_{2n}}\right) \] where \(\{2\}^m\) means the argument 2 repeated \(m\) times. Sums of \(Z(\vec s)\) over some sets of \(\vec s\) are found explicitly, generalizing the formula for \(\zeta (\{ 3,1\}^n)\) conjectured by D. Zagier and proved by J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek [Trans. Am. Math. Soc. 353, 907-941 (2001; Zbl 1002.11093)].
The technique is based on further development of the algebraic and combinatorial theory of shuffles introduced by K.-T. Chen [Proc. Lond. Math. Soc. (3) 4, 502-512 (1954; Zbl 0058.25603)] and R. Ree [Ann. Math. (2) 68, 210-220 (1958; Zbl 0083.25401)].

11M41 Other Dirichlet series and zeta functions
05A19 Combinatorial identities, bijective combinatorics
11B75 Other combinatorial number theory
Full Text: DOI arXiv
[1] 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, #R5, (1997) · Zbl 0884.40004
[2] Borwein, J.M.; Bradley, D.M.; Broadhurst, D.J.; Lisoněk, P., Special values of multiple polylogarithms, Trans. amer. math. soc., 353, 907-941, (2001) · Zbl 1002.11093
[3] Borwein, J.M.; Bradley, D.M.; Broadhurst, D.J.; Lisoněk, Petr, Combinatorial aspects of multiple zeta values, Electron. J. combin., 5, #R38, (1998) · Zbl 0904.05012
[4] D. Bowman, and, D. M. Bradley, Resolution of some open problems concerning multiple zeta evaluations of arbitrary depth, submitted for publication. · Zbl 1035.11037
[5] D. J. Broadhurst, private e-mail, 1997.
[6] Chen, K.-T., Iterated integrals and exponential homomorphisms, Proc. London math. soc. (3), 4, 502-512, (1954) · Zbl 0058.25603
[7] Chen, K.-T., Integration of paths, geometric invariants and a generalized baker – hausdorff formula, Ann. math., 65, 163-178, (1957) · Zbl 0077.25301
[8] Chen, K.-T., Algebras of iterated path integrals and fundamental groups, Trans. amer. math. soc., 156, 359-379, (1971) · Zbl 0217.47705
[9] Friedrichs, K.O., Mathematical aspects of the quantum theory of fields, V, Comm. pure appl. math., 6, 1-72, (1953) · Zbl 0052.44504
[10] Goncharov, A.B., Multiple polylogarithms, cyclotomy and modular complexes, Math. res. lett., 5, 497-516, (1998) · Zbl 0961.11040
[11] M. E. Hoffman, Algebraic structures on the set of multiple zeta values, preprint.
[12] M. E. Hoffman, and, Y. Ohno, Relations of multiple zeta values and their algebraic expression, preprint. · Zbl 1139.11322
[13] Lyndon, R.C., A theorem of Friedrichs, Michigan. math. J., 3, 27-29, (1956) · Zbl 0070.03005
[14] Magnus, W., On the exponential solution of differential equations for a linear operation, Comm. pure appl. math., 7, 649-673, (1954) · Zbl 0056.34102
[15] Minh, H.N.; Petitot, M., Lyndon words, polylogarithms and the Riemann ζ function, Discrete math., 217, 273-292, (2000) · Zbl 0959.68144
[16] 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
[17] Radford, D.E., A natural ring basis for the shuffle algebra and an application to group schemes, J. algebra, 58, 432-454, (1979) · Zbl 0409.16011
[18] Ree, R., Lie elements and an algebra associated with shuffles, Ann. math., 62, 210-220, (1958) · Zbl 0083.25401
[19] Zagier, D., Values of zeta functions and their applications, First European congress of mathematics, (1994), Birkhäuser Boston, p. 497-512 · Zbl 0822.11001
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