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Some multi-set inclusions associated with shuffle convolutions and multiple zeta values. (English) Zbl 1016.11035

The authors develop a new method for obtaining combinatorial identities involving shuffle convolutions [see D. Bowman and D. M. Bradley, J. Comb. Theory, Ser. A 97, 43-61 (2002; Zbl 1021.11026)]. As an application, a new proof of the formula \(\zeta (3,1,3,1,\ldots ,3,1)=2\pi^{4n}/(4n+2)!\), where \(\{ 3,1\}\) is repeated \(n\) times, is given. Some new identities for the multiple zeta function are also obtained.

MSC:

11M41 Other Dirichlet series and zeta functions
05A99 Enumerative combinatorics
05E99 Algebraic combinatorics

Citations:

Zbl 1021.11026
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Full Text: DOI

References:

[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, 2, #R5 (1997) · Zbl 0884.40004
[2] Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; Lisonĕk, P., Special values of multiple polylogarithms, Trans. Am. Math. Soc., 353, 3, 907-941 (2000) · Zbl 1002.11093
[3] Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; Lisonĕk, P., Combinatorial aspects of multiple zeta values, Electron. J. Combin., 5, 1, #R38 (1998) · Zbl 0904.05012
[4] D. Bowman, D.M. Bradley, Resolution of some open problems concerning multiple zeta evaluations of arbitrary depth, Compositio Math. (in press); D. Bowman, D.M. Bradley, Resolution of some open problems concerning multiple zeta evaluations of arbitrary depth, Compositio Math. (in press) · Zbl 1035.11037
[5] Bowman, D.; Bradley, D. M., The algebra and combinatorics of shuffles and multiple zeta values, J. Combin. Theory Ser. A, 97, 1, 43-61 (2002) · Zbl 1021.11026
[6] D. Bowman, D.M. Bradley, Multiple polylogarithms: a brief survey, Proceedings of a Conference on \(q\); D. Bowman, D.M. Bradley, Multiple polylogarithms: a brief survey, Proceedings of a Conference on \(q\) · Zbl 0998.33013
[7] Broadhurst, D. J.; Kreimer, D., Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B, 393, 3-4, 403-412 (1997) · Zbl 0946.81028
[8] Chen, Kuo-Tsai, Iterated integrals and exponential homomorphisms, Proc. London Math. Soc., 4, 3, 502-512 (1954) · Zbl 0058.25603
[9] Chen, Kuo-Tsai, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. Math., 65, 1, 163-178 (1957) · Zbl 0077.25301
[10] Goncharov, A. B., Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett., 5, 4, 497-516 (1998) · Zbl 0961.11040
[11] Hoffman, M. E., Multiple harmonic series, Pacific J. Math., 152, 2, 275-290 (1992) · Zbl 0763.11037
[12] Hoffman, M. E., The algebra of multiple harmonic series, J. Algebra, 194, 477-495 (1997) · Zbl 0881.11067
[13] Minh, Hoang Ngoc; Petitot, M., Lyndon words, polylogarithms and the Riemann \(ζ\) function, Discrete Math., 217, 1-3, 273-292 (2000) · Zbl 0959.68144
[14] 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
[15] Ree, R., Lie elements and an algebra associated with shuffles, Ann. Math., 62, 2, 210-220 (1958) · Zbl 0083.25401
[16] Ji Hoon Ryoo, Identities for multiple zeta values using the shuffle operation, Master’s Thesis, University of Maine, May 2001; Ji Hoon Ryoo, Identities for multiple zeta values using the shuffle operation, Master’s Thesis, University of Maine, May 2001
[17] Waldschmidt, M., Valeurs zêta multiples: une introduction, J. Théor. Nombres Bordeaux, 12, 2, 581-595 (2000) · Zbl 0976.11037
[18] M. Waldschmidt, Introduction to polylogarithms, Proceedings of the Chandigarh International Conference on Number Theory and Discrete Mathematics in Honour of Srinivasa Ramanujan (to appear); M. Waldschmidt, Introduction to polylogarithms, Proceedings of the Chandigarh International Conference on Number Theory and Discrete Mathematics in Honour of Srinivasa Ramanujan (to appear) · Zbl 1035.11033
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